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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  ontowfo 6326  cfv 6328  (class class class)co 7139  ωcom 7564  cen 8493  cdom 8494  csdm 8495  Fincfn 8496  1c1 10531   + caddc 10533  cn 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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