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Theorem 1stcfb 21528
 Description: For any point 𝐴 in a first-countable topology, there is a function 𝑓:ℕ⟶𝐽 enumerating neighborhoods of 𝐴 which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
1stcclb.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcfb ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))
Distinct variable groups:   𝑓,𝑘,𝑦,𝐴   𝑓,𝐽,𝑘,𝑦   𝑘,𝑋,𝑦
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem 1stcfb
Dummy variables 𝑎 𝑔 𝑛 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . 3 𝑋 = 𝐽
211stcclb 21527 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))
3 simplr 785 . . . . . . . . 9 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝐴𝑋)
4 eleq2 2833 . . . . . . . . . . 11 (𝑧 = 𝑋 → (𝐴𝑧𝐴𝑋))
5 sseq2 3787 . . . . . . . . . . . . 13 (𝑧 = 𝑋 → (𝑤𝑧𝑤𝑋))
65anbi2d 622 . . . . . . . . . . . 12 (𝑧 = 𝑋 → ((𝐴𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝑋)))
76rexbidv 3199 . . . . . . . . . . 11 (𝑧 = 𝑋 → (∃𝑤𝑥 (𝐴𝑤𝑤𝑧) ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑋)))
84, 7imbi12d 335 . . . . . . . . . 10 (𝑧 = 𝑋 → ((𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ↔ (𝐴𝑋 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑋))))
9 simprrr 800 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)))
10 1stctop 21526 . . . . . . . . . . . 12 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
1110ad2antrr 717 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝐽 ∈ Top)
121topopn 20990 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑋𝐽)
148, 9, 13rspcdva 3467 . . . . . . . . 9 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → (𝐴𝑋 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑋)))
153, 14mpd 15 . . . . . . . 8 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑤𝑥 (𝐴𝑤𝑤𝑋))
16 simpl 474 . . . . . . . . 9 ((𝐴𝑤𝑤𝑋) → 𝐴𝑤)
1716reximi 3157 . . . . . . . 8 (∃𝑤𝑥 (𝐴𝑤𝑤𝑋) → ∃𝑤𝑥 𝐴𝑤)
1815, 17syl 17 . . . . . . 7 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑤𝑥 𝐴𝑤)
19 eleq2w 2828 . . . . . . . 8 (𝑤 = 𝑎 → (𝐴𝑤𝐴𝑎))
2019cbvrexv 3320 . . . . . . 7 (∃𝑤𝑥 𝐴𝑤 ↔ ∃𝑎𝑥 𝐴𝑎)
2118, 20sylib 209 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑎𝑥 𝐴𝑎)
22 rabn0 4122 . . . . . 6 ({𝑎𝑥𝐴𝑎} ≠ ∅ ↔ ∃𝑎𝑥 𝐴𝑎)
2321, 22sylibr 225 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → {𝑎𝑥𝐴𝑎} ≠ ∅)
24 vex 3353 . . . . . . 7 𝑥 ∈ V
2524rabex 4973 . . . . . 6 {𝑎𝑥𝐴𝑎} ∈ V
26250sdom 8298 . . . . 5 (∅ ≺ {𝑎𝑥𝐴𝑎} ↔ {𝑎𝑥𝐴𝑎} ≠ ∅)
2723, 26sylibr 225 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∅ ≺ {𝑎𝑥𝐴𝑎})
28 ssrab2 3847 . . . . . 6 {𝑎𝑥𝐴𝑎} ⊆ 𝑥
29 ssdomg 8206 . . . . . 6 (𝑥 ∈ V → ({𝑎𝑥𝐴𝑎} ⊆ 𝑥 → {𝑎𝑥𝐴𝑎} ≼ 𝑥))
3024, 28, 29mp2 9 . . . . 5 {𝑎𝑥𝐴𝑎} ≼ 𝑥
31 simprrl 799 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑥 ≼ ω)
32 nnenom 12987 . . . . . . 7 ℕ ≈ ω
3332ensymi 8210 . . . . . 6 ω ≈ ℕ
34 domentr 8219 . . . . . 6 ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ)
3531, 33, 34sylancl 580 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑥 ≼ ℕ)
36 domtr 8213 . . . . 5 (({𝑎𝑥𝐴𝑎} ≼ 𝑥𝑥 ≼ ℕ) → {𝑎𝑥𝐴𝑎} ≼ ℕ)
3730, 35, 36sylancr 581 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → {𝑎𝑥𝐴𝑎} ≼ ℕ)
38 fodomr 8318 . . . 4 ((∅ ≺ {𝑎𝑥𝐴𝑎} ∧ {𝑎𝑥𝐴𝑎} ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})
3927, 37, 38syl2anc 579 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑔 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})
4010ad3antrrr 721 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Top)
41 imassrn 5659 . . . . . . . . . 10 (𝑔 “ (1...𝑛)) ⊆ ran 𝑔
42 forn 6301 . . . . . . . . . . . . 13 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → ran 𝑔 = {𝑎𝑥𝐴𝑎})
4342ad2antll 720 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ran 𝑔 = {𝑎𝑥𝐴𝑎})
44 simprll 797 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑥 ∈ 𝒫 𝐽)
4544elpwid 4327 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑥𝐽)
4628, 45syl5ss 3772 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → {𝑎𝑥𝐴𝑎} ⊆ 𝐽)
4743, 46eqsstrd 3799 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ran 𝑔𝐽)
4847adantr 472 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → ran 𝑔𝐽)
4941, 48syl5ss 3772 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ⊆ 𝐽)
50 elfznn 12577 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
5150ssriv 3765 . . . . . . . . . . . . . 14 (1...𝑛) ⊆ ℕ
52 fof 6298 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → 𝑔:ℕ⟶{𝑎𝑥𝐴𝑎})
5352ad2antll 720 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑔:ℕ⟶{𝑎𝑥𝐴𝑎})
5453fdmd 6232 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → dom 𝑔 = ℕ)
5551, 54syl5sseqr 3814 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (1...𝑛) ⊆ dom 𝑔)
5655adantr 472 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ dom 𝑔)
57 sseqin2 3979 . . . . . . . . . . . 12 ((1...𝑛) ⊆ dom 𝑔 ↔ (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
5856, 57sylib 209 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
59 elfz1end 12578 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (1...𝑛))
60 ne0i 4085 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑛) → (1...𝑛) ≠ ∅)
6160adantl 473 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ (1...𝑛)) → (1...𝑛) ≠ ∅)
6259, 61sylan2b 587 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ≠ ∅)
6358, 62eqnetrd 3004 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
64 imadisj 5666 . . . . . . . . . . 11 ((𝑔 “ (1...𝑛)) = ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) = ∅)
6564necon3bii 2989 . . . . . . . . . 10 ((𝑔 “ (1...𝑛)) ≠ ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
6663, 65sylibr 225 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ≠ ∅)
67 fzfid 12980 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
6853ffund 6227 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → Fun 𝑔)
6968adantr 472 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → Fun 𝑔)
70 fores 6307 . . . . . . . . . . 11 ((Fun 𝑔 ∧ (1...𝑛) ⊆ dom 𝑔) → (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛)))
7169, 56, 70syl2anc 579 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛)))
72 fofi 8459 . . . . . . . . . 10 (((1...𝑛) ∈ Fin ∧ (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛))) → (𝑔 “ (1...𝑛)) ∈ Fin)
7367, 71, 72syl2anc 579 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ Fin)
74 fiinopn 20985 . . . . . . . . . 10 (𝐽 ∈ Top → (((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin) → (𝑔 “ (1...𝑛)) ∈ 𝐽))
7574imp 395 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin)) → (𝑔 “ (1...𝑛)) ∈ 𝐽)
7640, 49, 66, 73, 75syl13anc 1491 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ 𝐽)
7776fmpttd 6575 . . . . . . 7 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽)
78 imassrn 5659 . . . . . . . . . . . . 13 (𝑔 “ (1...𝑘)) ⊆ ran 𝑔
7943adantr 472 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ran 𝑔 = {𝑎𝑥𝐴𝑎})
8078, 79syl5sseq 3813 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ {𝑎𝑥𝐴𝑎})
81 id 22 . . . . . . . . . . . . . 14 (𝐴𝑛𝐴𝑛)
8281rgenw 3071 . . . . . . . . . . . . 13 𝑛𝑥 (𝐴𝑛𝐴𝑛)
83 eleq2w 2828 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝐴𝑎𝐴𝑛))
8483ralrab 3525 . . . . . . . . . . . . 13 (∀𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛 ↔ ∀𝑛𝑥 (𝐴𝑛𝐴𝑛))
8582, 84mpbir 222 . . . . . . . . . . . 12 𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛
86 ssralv 3826 . . . . . . . . . . . 12 ((𝑔 “ (1...𝑘)) ⊆ {𝑎𝑥𝐴𝑎} → (∀𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛 → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛))
8780, 85, 86mpisyl 21 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛)
88 elintg 4641 . . . . . . . . . . . 12 (𝐴𝑋 → (𝐴 (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛))
8988ad3antlr 722 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛))
9087, 89mpbird 248 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 (𝑔 “ (1...𝑘)))
91 simpr 477 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
9276ralrimiva 3113 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽)
93 oveq2 6850 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
9493imaeq2d 5648 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘)))
9594inteqd 4638 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘)))
9695eleq1d 2829 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ( (𝑔 “ (1...𝑛)) ∈ 𝐽 (𝑔 “ (1...𝑘)) ∈ 𝐽))
9796rspccva 3460 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ∈ 𝐽)
9892, 97sylan 575 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ∈ 𝐽)
99 eqid 2765 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))
10095, 99fvmptg 6469 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ (𝑔 “ (1...𝑘)) ∈ 𝐽) → ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) = (𝑔 “ (1...𝑘)))
10191, 98, 100syl2anc 579 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) = (𝑔 “ (1...𝑘)))
10290, 101eleqtrrd 2847 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))
103 fzssp1 12591 . . . . . . . . . . . 12 (1...𝑘) ⊆ (1...(𝑘 + 1))
104 imass2 5683 . . . . . . . . . . . 12 ((1...𝑘) ⊆ (1...(𝑘 + 1)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))))
105103, 104mp1i 13 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))))
106 intss 4654 . . . . . . . . . . 11 ((𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))) → (𝑔 “ (1...(𝑘 + 1))) ⊆ (𝑔 “ (1...𝑘)))
107105, 106syl 17 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...(𝑘 + 1))) ⊆ (𝑔 “ (1...𝑘)))
108 peano2nn 11288 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
109108adantl 473 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
110 oveq2 6850 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
111110imaeq2d 5648 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1))))
112111inteqd 4638 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1))))
113112eleq1d 2829 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ( (𝑔 “ (1...𝑛)) ∈ 𝐽 (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽))
114113rspccva 3460 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ (𝑘 + 1) ∈ ℕ) → (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
11592, 108, 114syl2an 589 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
116112, 99fvmptg 6469 . . . . . . . . . . 11 (((𝑘 + 1) ∈ ℕ ∧ (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽) → ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) = (𝑔 “ (1...(𝑘 + 1))))
117109, 115, 116syl2anc 579 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) = (𝑔 “ (1...(𝑘 + 1))))
118107, 117, 1013sstr4d 3808 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))
119102, 118jca 507 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
120119ralrimiva 3113 . . . . . . 7 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
121 simprlr 798 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)))
122 eleq2w 2828 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝐴𝑧𝐴𝑦))
123 sseq2 3787 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑤𝑧𝑤𝑦))
124123anbi2d 622 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐴𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝑦)))
125124rexbidv 3199 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (∃𝑤𝑥 (𝐴𝑤𝑤𝑧) ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
126122, 125imbi12d 335 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ↔ (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦))))
127126rspccva 3460 . . . . . . . . . . 11 ((∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
128121, 127sylan 575 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
129 eleq2w 2828 . . . . . . . . . . . 12 (𝑎 = 𝑤 → (𝐴𝑎𝐴𝑤))
130129rexrab 3527 . . . . . . . . . . 11 (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑦))
13143rexeqdv 3293 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦))
132 fofn 6300 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → 𝑔 Fn ℕ)
133132ad2antll 720 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑔 Fn ℕ)
134 sseq1 3786 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑔𝑘) → (𝑤𝑦 ↔ (𝑔𝑘) ⊆ 𝑦))
135134rexrn 6551 . . . . . . . . . . . . . . 15 (𝑔 Fn ℕ → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
136133, 135syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
137131, 136bitr3d 272 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
138137adantr 472 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
139 elfz1end 12578 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (1...𝑘))
14068adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → Fun 𝑔)
141 elfznn 12577 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
142141ssriv 3765 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
14354adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → dom 𝑔 = ℕ)
144142, 143syl5sseqr 3814 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (1...𝑘) ⊆ dom 𝑔)
145 funfvima2 6686 . . . . . . . . . . . . . . . . 17 ((Fun 𝑔 ∧ (1...𝑘) ⊆ dom 𝑔) → (𝑘 ∈ (1...𝑘) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘))))
146140, 144, 145syl2anc 579 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝑘 ∈ (1...𝑘) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘))))
147146imp 395 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) ∧ 𝑘 ∈ (1...𝑘)) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘)))
148139, 147sylan2b 587 . . . . . . . . . . . . . 14 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘)))
149 intss1 4648 . . . . . . . . . . . . . 14 ((𝑔𝑘) ∈ (𝑔 “ (1...𝑘)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔𝑘))
150 sstr2 3768 . . . . . . . . . . . . . 14 ( (𝑔 “ (1...𝑘)) ⊆ (𝑔𝑘) → ((𝑔𝑘) ⊆ 𝑦 (𝑔 “ (1...𝑘)) ⊆ 𝑦))
151148, 149, 1503syl 18 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑔𝑘) ⊆ 𝑦 (𝑔 “ (1...𝑘)) ⊆ 𝑦))
152151reximdva 3163 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
153138, 152sylbid 231 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
154130, 153syl5bir 234 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑤𝑥 (𝐴𝑤𝑤𝑦) → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
155128, 154syld 47 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
156101sseq1d 3792 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 (𝑔 “ (1...𝑘)) ⊆ 𝑦))
157156rexbidva 3196 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
158157adantr 472 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
159155, 158sylibrd 250 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
160159ralrimiva 3113 . . . . . . 7 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
161 nnex 11281 . . . . . . . . 9 ℕ ∈ V
162161mptex 6679 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) ∈ V
163 feq1 6204 . . . . . . . . 9 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓:ℕ⟶𝐽 ↔ (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽))
164 fveq1 6374 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓𝑘) = ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))
165164eleq2d 2830 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝐴 ∈ (𝑓𝑘) ↔ 𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
166 fveq1 6374 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)))
167166, 164sseq12d 3794 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘) ↔ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
168165, 167anbi12d 624 . . . . . . . . . 10 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ↔ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))))
169168ralbidv 3133 . . . . . . . . 9 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ↔ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))))
170164sseq1d 3792 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝑓𝑘) ⊆ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
171170rexbidv 3199 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
172171imbi2d 331 . . . . . . . . . 10 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦) ↔ (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
173172ralbidv 3133 . . . . . . . . 9 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
174163, 169, 1733anbi123d 1560 . . . . . . . 8 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)) ↔ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))))
175162, 174spcev 3452 . . . . . . 7 (((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))
17677, 120, 160, 175syl3anc 1490 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))
177176expr 448 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)))) → (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦))))
178177adantrrl 715 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦))))
179178exlimdv 2028 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → (∃𝑔 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦))))
18039, 179mpd 15 . 2 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))
1812, 180rexlimddv 3182 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652  ∃wex 1874   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  {crab 3059  Vcvv 3350   ∩ cin 3731   ⊆ wss 3732  ∅c0 4079  𝒫 cpw 4315  ∪ cuni 4594  ∩ cint 4633   class class class wbr 4809   ↦ cmpt 4888  dom cdm 5277  ran crn 5278   ↾ cres 5279   “ cima 5280  Fun wfun 6062   Fn wfn 6063  ⟶wf 6064  –onto→wfo 6066  ‘cfv 6068  (class class class)co 6842  ωcom 7263   ≈ cen 8157   ≼ cdom 8158   ≺ csdm 8159  Fincfn 8160  1c1 10190   + caddc 10192  ℕcn 11274  ...cfz 12533  Topctop 20977  1st𝜔c1stc 21520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-top 20978  df-1stc 21522 This theorem is referenced by:  1stcelcls  21544
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