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Theorem 1stcfb 22054
 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 22053 . 2 ((𝐽 ∈ 1stω ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))
3 simplr 768 . . . . . . . . 9 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝐴𝑋)
4 eleq2 2881 . . . . . . . . . . 11 (𝑧 = 𝑋 → (𝐴𝑧𝐴𝑋))
5 sseq2 3944 . . . . . . . . . . . . 13 (𝑧 = 𝑋 → (𝑤𝑧𝑤𝑋))
65anbi2d 631 . . . . . . . . . . . 12 (𝑧 = 𝑋 → ((𝐴𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝑋)))
76rexbidv 3259 . . . . . . . . . . 11 (𝑧 = 𝑋 → (∃𝑤𝑥 (𝐴𝑤𝑤𝑧) ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑋)))
84, 7imbi12d 348 . . . . . . . . . 10 (𝑧 = 𝑋 → ((𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ↔ (𝐴𝑋 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑋))))
9 simprrr 781 . . . . . . . . . 10 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)))
10 1stctop 22052 . . . . . . . . . . . 12 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
1110ad2antrr 725 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝐽 ∈ Top)
121topopn 21515 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . . . . 10 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑋𝐽)
148, 9, 13rspcdva 3576 . . . . . . . . 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 486 . . . . . . . . 9 ((𝐴𝑤𝑤𝑋) → 𝐴𝑤)
1716reximi 3209 . . . . . . . 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 2876 . . . . . . . 8 (𝑤 = 𝑎 → (𝐴𝑤𝐴𝑎))
2019cbvrexvw 3400 . . . . . . 7 (∃𝑤𝑥 𝐴𝑤 ↔ ∃𝑎𝑥 𝐴𝑎)
2118, 20sylib 221 . . . . . 6 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑎𝑥 𝐴𝑎)
22 rabn0 4296 . . . . . 6 ({𝑎𝑥𝐴𝑎} ≠ ∅ ↔ ∃𝑎𝑥 𝐴𝑎)
2321, 22sylibr 237 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → {𝑎𝑥𝐴𝑎} ≠ ∅)
24 vex 3447 . . . . . . 7 𝑥 ∈ V
2524rabex 5202 . . . . . 6 {𝑎𝑥𝐴𝑎} ∈ V
26250sdom 8636 . . . . 5 (∅ ≺ {𝑎𝑥𝐴𝑎} ↔ {𝑎𝑥𝐴𝑎} ≠ ∅)
2723, 26sylibr 237 . . . 4 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∅ ≺ {𝑎𝑥𝐴𝑎})
28 ssrab2 4010 . . . . . 6 {𝑎𝑥𝐴𝑎} ⊆ 𝑥
29 ssdomg 8542 . . . . . 6 (𝑥 ∈ V → ({𝑎𝑥𝐴𝑎} ⊆ 𝑥 → {𝑎𝑥𝐴𝑎} ≼ 𝑥))
3024, 28, 29mp2 9 . . . . 5 {𝑎𝑥𝐴𝑎} ≼ 𝑥
31 simprrl 780 . . . . . 6 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑥 ≼ ω)
32 nnenom 13347 . . . . . . 7 ℕ ≈ ω
3332ensymi 8546 . . . . . 6 ω ≈ ℕ
34 domentr 8555 . . . . . 6 ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ)
3531, 33, 34sylancl 589 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → 𝑥 ≼ ℕ)
36 domtr 8549 . . . . 5 (({𝑎𝑥𝐴𝑎} ≼ 𝑥𝑥 ≼ ℕ) → {𝑎𝑥𝐴𝑎} ≼ ℕ)
3730, 35, 36sylancr 590 . . . 4 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → {𝑎𝑥𝐴𝑎} ≼ ℕ)
38 fodomr 8656 . . . 4 ((∅ ≺ {𝑎𝑥𝐴𝑎} ∧ {𝑎𝑥𝐴𝑎} ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})
3927, 37, 38syl2anc 587 . . 3 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))))) → ∃𝑔 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})
4010ad3antrrr 729 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Top)
41 imassrn 5911 . . . . . . . . . 10 (𝑔 “ (1...𝑛)) ⊆ ran 𝑔
42 forn 6572 . . . . . . . . . . . . 13 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → ran 𝑔 = {𝑎𝑥𝐴𝑎})
4342ad2antll 728 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ran 𝑔 = {𝑎𝑥𝐴𝑎})
44 simprll 778 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑥 ∈ 𝒫 𝐽)
4544elpwid 4511 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑥𝐽)
4628, 45sstrid 3929 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → {𝑎𝑥𝐴𝑎} ⊆ 𝐽)
4743, 46eqsstrd 3956 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ran 𝑔𝐽)
4847adantr 484 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → ran 𝑔𝐽)
4941, 48sstrid 3929 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ⊆ 𝐽)
50 fz1ssnn 12937 . . . . . . . . . . . . . 14 (1...𝑛) ⊆ ℕ
51 fof 6569 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → 𝑔:ℕ⟶{𝑎𝑥𝐴𝑎})
5251ad2antll 728 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑔:ℕ⟶{𝑎𝑥𝐴𝑎})
5352fdmd 6501 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → dom 𝑔 = ℕ)
5450, 53sseqtrrid 3971 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (1...𝑛) ⊆ dom 𝑔)
5554adantr 484 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ dom 𝑔)
56 sseqin2 4145 . . . . . . . . . . . 12 ((1...𝑛) ⊆ dom 𝑔 ↔ (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
5755, 56sylib 221 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
58 elfz1end 12936 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (1...𝑛))
59 ne0i 4253 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑛) → (1...𝑛) ≠ ∅)
6059adantl 485 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ (1...𝑛)) → (1...𝑛) ≠ ∅)
6158, 60sylan2b 596 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ≠ ∅)
6257, 61eqnetrd 3057 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
63 imadisj 5919 . . . . . . . . . . 11 ((𝑔 “ (1...𝑛)) = ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) = ∅)
6463necon3bii 3042 . . . . . . . . . 10 ((𝑔 “ (1...𝑛)) ≠ ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
6562, 64sylibr 237 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ≠ ∅)
66 fzfid 13340 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
6752ffund 6495 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → Fun 𝑔)
68 fores 6579 . . . . . . . . . . 11 ((Fun 𝑔 ∧ (1...𝑛) ⊆ dom 𝑔) → (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛)))
6967, 55, 68syl2an2r 684 . . . . . . . . . 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)))
70 fofi 8798 . . . . . . . . . 10 (((1...𝑛) ∈ Fin ∧ (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛))) → (𝑔 “ (1...𝑛)) ∈ Fin)
7166, 69, 70syl2anc 587 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ Fin)
72 fiinopn 21510 . . . . . . . . . 10 (𝐽 ∈ Top → (((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin) → (𝑔 “ (1...𝑛)) ∈ 𝐽))
7372imp 410 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin)) → (𝑔 “ (1...𝑛)) ∈ 𝐽)
7440, 49, 65, 71, 73syl13anc 1369 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ 𝐽)
7574fmpttd 6860 . . . . . . 7 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽)
76 imassrn 5911 . . . . . . . . . . . . 13 (𝑔 “ (1...𝑘)) ⊆ ran 𝑔
7743adantr 484 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ran 𝑔 = {𝑎𝑥𝐴𝑎})
7876, 77sseqtrid 3970 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ {𝑎𝑥𝐴𝑎})
79 id 22 . . . . . . . . . . . . . 14 (𝐴𝑛𝐴𝑛)
8079rgenw 3121 . . . . . . . . . . . . 13 𝑛𝑥 (𝐴𝑛𝐴𝑛)
81 eleq2w 2876 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝐴𝑎𝐴𝑛))
8281ralrab 3636 . . . . . . . . . . . . 13 (∀𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛 ↔ ∀𝑛𝑥 (𝐴𝑛𝐴𝑛))
8380, 82mpbir 234 . . . . . . . . . . . 12 𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛
84 ssralv 3984 . . . . . . . . . . . 12 ((𝑔 “ (1...𝑘)) ⊆ {𝑎𝑥𝐴𝑎} → (∀𝑛 ∈ {𝑎𝑥𝐴𝑎}𝐴𝑛 → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛))
8578, 83, 84mpisyl 21 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛)
86 elintg 4849 . . . . . . . . . . . 12 (𝐴𝑋 → (𝐴 (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴𝑛))
8786ad3antlr 730 . . . . . . . . . . 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))
8885, 87mpbird 260 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 (𝑔 “ (1...𝑘)))
89 eqid 2801 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))
90 oveq2 7147 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
9190imaeq2d 5900 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘)))
9291inteqd 4846 . . . . . . . . . . 11 (𝑛 = 𝑘 (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘)))
93 simpr 488 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
9474ralrimiva 3152 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽)
9592eleq1d 2877 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ( (𝑔 “ (1...𝑛)) ∈ 𝐽 (𝑔 “ (1...𝑘)) ∈ 𝐽))
9695rspccva 3573 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ∈ 𝐽)
9794, 96sylan 583 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ∈ 𝐽)
9889, 92, 93, 97fvmptd3 6772 . . . . . . . . . 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)))
9988, 98eleqtrrd 2896 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))
100 fzssp1 12949 . . . . . . . . . . . 12 (1...𝑘) ⊆ (1...(𝑘 + 1))
101 imass2 5936 . . . . . . . . . . . 12 ((1...𝑘) ⊆ (1...(𝑘 + 1)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))))
102100, 101mp1i 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))))
103 intss 4862 . . . . . . . . . . 11 ((𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))) → (𝑔 “ (1...(𝑘 + 1))) ⊆ (𝑔 “ (1...𝑘)))
104102, 103syl 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)))
105 oveq2 7147 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
106105imaeq2d 5900 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1))))
107106inteqd 4846 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1))))
108 peano2nn 11641 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
109108adantl 485 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
110107eleq1d 2877 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ( (𝑔 “ (1...𝑛)) ∈ 𝐽 (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽))
111110rspccva 3573 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ (𝑘 + 1) ∈ ℕ) → (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
11294, 108, 111syl2an 598 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
11389, 107, 109, 112fvmptd3 6772 . . . . . . . . . 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))))
114104, 113, 983sstr4d 3965 . . . . . . . . 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))
11599, 114jca 515 . . . . . . . 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)))
116115ralrimiva 3152 . . . . . . 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)))
117 simprlr 779 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)))
118 eleq2w 2876 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝐴𝑧𝐴𝑦))
119 sseq2 3944 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑤𝑧𝑤𝑦))
120119anbi2d 631 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐴𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝑦)))
121120rexbidv 3259 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (∃𝑤𝑥 (𝐴𝑤𝑤𝑧) ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
122118, 121imbi12d 348 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ↔ (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦))))
123122rspccva 3573 . . . . . . . . . . 11 ((∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧)) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
124117, 123sylan 583 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑦)))
125 eleq2w 2876 . . . . . . . . . . . 12 (𝑎 = 𝑤 → (𝐴𝑎𝐴𝑤))
126125rexrab 3638 . . . . . . . . . . 11 (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑤𝑥 (𝐴𝑤𝑤𝑦))
12743rexeqdv 3368 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦))
128 fofn 6571 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎} → 𝑔 Fn ℕ)
129128ad2antll 728 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → 𝑔 Fn ℕ)
130 sseq1 3943 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑔𝑘) → (𝑤𝑦 ↔ (𝑔𝑘) ⊆ 𝑦))
131130rexrn 6834 . . . . . . . . . . . . . . 15 (𝑔 Fn ℕ → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
132129, 131syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
133127, 132bitr3d 284 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) → (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
134133adantr 484 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑤 ∈ {𝑎𝑥𝐴𝑎}𝑤𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑦))
135 elfz1end 12936 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (1...𝑘))
136 fz1ssnn 12937 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
13753adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → dom 𝑔 = ℕ)
138136, 137sseqtrrid 3971 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (1...𝑘) ⊆ dom 𝑔)
139 funfvima2 6975 . . . . . . . . . . . . . . . . 17 ((Fun 𝑔 ∧ (1...𝑘) ⊆ dom 𝑔) → (𝑘 ∈ (1...𝑘) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘))))
14067, 138, 139syl2an2r 684 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝑘 ∈ (1...𝑘) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘))))
141140imp 410 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) ∧ 𝑘 ∈ (1...𝑘)) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘)))
142135, 141sylan2b 596 . . . . . . . . . . . . . 14 (((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ (𝑔 “ (1...𝑘)))
143 intss1 4856 . . . . . . . . . . . . . 14 ((𝑔𝑘) ∈ (𝑔 “ (1...𝑘)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔𝑘))
144 sstr2 3925 . . . . . . . . . . . . . 14 ( (𝑔 “ (1...𝑘)) ⊆ (𝑔𝑘) → ((𝑔𝑘) ⊆ 𝑦 (𝑔 “ (1...𝑘)) ⊆ 𝑦))
145142, 143, 1443syl 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))
146145reximdva 3236 . . . . . . . . . . . 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))
147134, 146sylbid 243 . . . . . . . . . . 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))
148126, 147syl5bir 246 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (∃𝑤𝑥 (𝐴𝑤𝑤𝑦) → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
149124, 148syld 47 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧𝐽 (𝐴𝑧 → ∃𝑤𝑥 (𝐴𝑤𝑤𝑧))) ∧ 𝑔:ℕ–onto→{𝑎𝑥𝐴𝑎})) ∧ 𝑦𝐽) → (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
15098sseq1d 3949 . . . . . . . . . . 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))
151150rexbidva 3258 . . . . . . . . . 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))
152151adantr 484 . . . . . . . . 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))
153149, 152sylibrd 262 . . . . . . . 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))
154153ralrimiva 3152 . . . . . . 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))
155 nnex 11635 . . . . . . . . 9 ℕ ∈ V
156155mptex 6967 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) ∈ V
157 feq1 6472 . . . . . . . . 9 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓:ℕ⟶𝐽 ↔ (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))):ℕ⟶𝐽))
158 fveq1 6648 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓𝑘) = ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘))
159158eleq2d 2878 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝐴 ∈ (𝑓𝑘) ↔ 𝐴 ∈ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
160 fveq1 6648 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (𝑓‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)))
161160, 158sseq12d 3951 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘) ↔ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘)))
162159, 161anbi12d 633 . . . . . . . . . 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))))
163162ralbidv 3165 . . . . . . . . 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))))
164158sseq1d 3949 . . . . . . . . . . . 12 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝑓𝑘) ⊆ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
165164rexbidv 3259 . . . . . . . . . . 11 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
166165imbi2d 344 . . . . . . . . . 10 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → ((𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦) ↔ (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
167166ralbidv 3165 . . . . . . . . 9 (𝑓 = (𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛))) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
168157, 163, 1673anbi123d 1433 . . . . . . . 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))))
169156, 168spcev 3558 . . . . . . 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)))
17075, 116, 154, 169syl3anc 1368 . . . . . 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)))
171170expr 460 . . . . 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))))
172171adantrrl 723 . . . 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))))
173172exlimdv 1934 . . 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))))
17439, 173mpd 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)))
1752, 174rexlimddv 3253 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884  ∅c0 4246  𝒫 cpw 4500  ∪ cuni 4803  ∩ cint 4841   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  ran crn 5524   ↾ cres 5525   “ cima 5526  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7139  ωcom 7564   ≈ cen 8493   ≼ cdom 8494   ≺ csdm 8495  Fincfn 8496  1c1 10531   + caddc 10533  ℕcn 11629  ...cfz 12889  Topctop 21502  1stωc1stc 22046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-top 21503  df-1stc 22048 This theorem is referenced by:  1stcelcls  22070
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