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Theorem alexsubALT 21765
Description: The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
alexsubALT.1 𝑋 = 𝐽
Assertion
Ref Expression
alexsubALT (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
Distinct variable groups:   𝑐,𝑑,𝑥,𝐽   𝑋,𝑐,𝑑,𝑥

Proof of Theorem alexsubALT
Dummy variables 𝑎 𝑏 𝑓 𝑡 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alexsubALT.1 . . 3 𝑋 = 𝐽
21alexsubALTlem1 21761 . 2 (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
31alexsubALTlem4 21764 . . . . 5 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) → ∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
4 selpw 4137 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
5 eleq2 2687 . . . . . . . . . . . . . . . . . . 19 (𝑋 = 𝑐 → (𝑡𝑋𝑡 𝑐))
653ad2ant3 1082 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋𝑡 𝑐))
7 eluni 4405 . . . . . . . . . . . . . . . . . . . 20 (𝑡 𝑐 ↔ ∃𝑤(𝑡𝑤𝑤𝑐))
8 ssel 3577 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐𝐽 → (𝑤𝑐𝑤𝐽))
9 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤𝐽𝑤 ∈ (topGen‘(fi‘𝑥))))
10 tg2 20680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑤 ∈ (topGen‘(fi‘𝑥)) ∧ 𝑡𝑤) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))
1110ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (topGen‘(fi‘𝑥)) → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤)))
129, 11syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤𝐽 → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))))
138, 12sylan9r 689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑤𝑐 → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))))
14133impia 1258 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤)))
15 sseq2 3606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑤 → (𝑦𝑧𝑦𝑤))
1615rspcev 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑤𝑐𝑦𝑤) → ∃𝑧𝑐 𝑦𝑧)
1716ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤𝑐 → (𝑦𝑤 → ∃𝑧𝑐 𝑦𝑧))
18173ad2ant3 1082 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (𝑦𝑤 → ∃𝑧𝑐 𝑦𝑧))
1918anim2d 588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → ((𝑡𝑦𝑦𝑤) → (𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
2019reximdv 3010 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
2114, 20syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
22213expia 1264 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑤𝑐 → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧))))
2322com23 86 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑡𝑤 → (𝑤𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧))))
2423impd 447 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → ((𝑡𝑤𝑤𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
2524exlimdv 1858 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (∃𝑤(𝑡𝑤𝑤𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
267, 25syl5bi 232 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑡 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
27263adant3 1079 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
286, 27sylbid 230 . . . . . . . . . . . . . . . . 17 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
29 ssel 3577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑧 → (𝑡𝑦𝑡𝑧))
30 elunii 4407 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡𝑧𝑧𝑐) → 𝑡 𝑐)
3130expcom 451 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑐 → (𝑡𝑧𝑡 𝑐))
326biimprd 238 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡 𝑐𝑡𝑋))
3331, 32sylan9r 689 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑧𝑐) → (𝑡𝑧𝑡𝑋))
3429, 33syl9r 78 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑧𝑐) → (𝑦𝑧 → (𝑡𝑦𝑡𝑋)))
3534rexlimdva 3024 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∃𝑧𝑐 𝑦𝑧 → (𝑡𝑦𝑡𝑋)))
3635com23 86 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑦 → (∃𝑧𝑐 𝑦𝑧𝑡𝑋)))
3736impd 447 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → ((𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧) → 𝑡𝑋))
3837rexlimdvw 3027 . . . . . . . . . . . . . . . . 17 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧) → 𝑡𝑋))
3928, 38impbid 202 . . . . . . . . . . . . . . . 16 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋 ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
40 elunirab 4414 . . . . . . . . . . . . . . . 16 (𝑡 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧))
4139, 40syl6bbr 278 . . . . . . . . . . . . . . 15 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋𝑡 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
4241eqrdv 2619 . . . . . . . . . . . . . 14 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → 𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
43 ssrab2 3666 . . . . . . . . . . . . . . . 16 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ⊆ (fi‘𝑥)
44 fvex 6158 . . . . . . . . . . . . . . . . 17 (fi‘𝑥) ∈ V
4544elpw2 4788 . . . . . . . . . . . . . . . 16 ({𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∈ 𝒫 (fi‘𝑥) ↔ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ⊆ (fi‘𝑥))
4643, 45mpbir 221 . . . . . . . . . . . . . . 15 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∈ 𝒫 (fi‘𝑥)
47 unieq 4410 . . . . . . . . . . . . . . . . . 18 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → 𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
4847eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑋 = 𝑎𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
49 pweq 4133 . . . . . . . . . . . . . . . . . . 19 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → 𝒫 𝑎 = 𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
5049ineq1d 3791 . . . . . . . . . . . . . . . . . 18 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝒫 𝑎 ∩ Fin) = (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin))
5150rexeqdv 3134 . . . . . . . . . . . . . . . . 17 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏 ↔ ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏))
5248, 51imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ((𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) ↔ (𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏)))
5352rspcv 3291 . . . . . . . . . . . . . . 15 ({𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∈ 𝒫 (fi‘𝑥) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏)))
5446, 53ax-mp 5 . . . . . . . . . . . . . 14 (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏))
5542, 54syl5com 31 . . . . . . . . . . . . 13 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏))
56 elfpw 8212 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin) ↔ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∧ 𝑏 ∈ Fin))
57 ssel 3577 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑡𝑏𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
58 sseq1 3605 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑡 → (𝑦𝑧𝑡𝑧))
5958rexbidv 3045 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑡 → (∃𝑧𝑐 𝑦𝑧 ↔ ∃𝑧𝑐 𝑡𝑧))
6059elrab 3346 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ↔ (𝑡 ∈ (fi‘𝑥) ∧ ∃𝑧𝑐 𝑡𝑧))
6160simprbi 480 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑧𝑐 𝑡𝑧)
6257, 61syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑡𝑏 → ∃𝑧𝑐 𝑡𝑧))
6362ralrimiv 2959 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∀𝑡𝑏𝑧𝑐 𝑡𝑧)
64 sseq2 3606 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝑓𝑡) → (𝑡𝑧𝑡 ⊆ (𝑓𝑡)))
6564ac6sfi 8148 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ Fin ∧ ∀𝑡𝑏𝑧𝑐 𝑡𝑧) → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)))
6665ex 450 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ Fin → (∀𝑡𝑏𝑧𝑐 𝑡𝑧 → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))))
6763, 66syl5 34 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ Fin → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))))
6867adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))))
69 simprll 801 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑓:𝑏𝑐)
70 frn 6010 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓:𝑏𝑐 → ran 𝑓𝑐)
7169, 70syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝑐)
72 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 ∈ Fin)
73 ffn 6002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:𝑏𝑐𝑓 Fn 𝑏)
74 dffn4 6078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 Fn 𝑏𝑓:𝑏onto→ran 𝑓)
7573, 74sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:𝑏𝑐𝑓:𝑏onto→ran 𝑓)
7675adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) → 𝑓:𝑏onto→ran 𝑓)
7776ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑓:𝑏onto→ran 𝑓)
78 fodomfi 8183 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏 ∈ Fin ∧ 𝑓:𝑏onto→ran 𝑓) → ran 𝑓𝑏)
7972, 77, 78syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝑏)
80 domfi 8125 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ Fin ∧ ran 𝑓𝑏) → ran 𝑓 ∈ Fin)
8172, 79, 80syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓 ∈ Fin)
8271, 81jca 554 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → (ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin))
83 elin 3774 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ↔ (ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin))
84 vex 3189 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐 ∈ V
8584elpw2 4788 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran 𝑓 ∈ 𝒫 𝑐 ↔ ran 𝑓𝑐)
8685anbi1i 730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin) ↔ (ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin))
8783, 86bitr2i 265 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin) ↔ ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
8882, 87sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
89 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 = 𝑏)
90 uniiun 4539 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑏 = 𝑡𝑏 𝑡
91 simprlr 802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))
92 ss2iun 4502 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡) → 𝑡𝑏 𝑡 𝑡𝑏 (𝑓𝑡))
9391, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑡𝑏 𝑡 𝑡𝑏 (𝑓𝑡))
9490, 93syl5eqss 3628 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 𝑡𝑏 (𝑓𝑡))
95 fniunfv 6459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑏 𝑡𝑏 (𝑓𝑡) = ran 𝑓)
9669, 73, 953syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑡𝑏 (𝑓𝑡) = ran 𝑓)
9794, 96sseqtrd 3620 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 ran 𝑓)
9889, 97eqsstrd 3618 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 ran 𝑓)
99 simpll2 1099 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑐𝐽)
10071, 99sstrd 3593 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝐽)
101 uniss 4424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran 𝑓𝐽 ran 𝑓 𝐽)
102101, 1syl6sseqr 3631 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑓𝐽 ran 𝑓𝑋)
103100, 102syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝑋)
10498, 103eqssd 3600 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 = ran 𝑓)
105 unieq 4410 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = ran 𝑓 𝑑 = ran 𝑓)
106105eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = ran 𝑓 → (𝑋 = 𝑑𝑋 = ran 𝑓))
107106rspcev 3295 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
10888, 104, 107syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
109108exp32 630 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
110109exlimdv 1858 . . . . . . . . . . . . . . . . . . 19 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → (∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
11168, 110syld 47 . . . . . . . . . . . . . . . . . 18 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
112111ex 450 . . . . . . . . . . . . . . . . 17 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑏 ∈ Fin → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
113112com23 86 . . . . . . . . . . . . . . . 16 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑏 ∈ Fin → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
114113impd 447 . . . . . . . . . . . . . . 15 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → ((𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∧ 𝑏 ∈ Fin) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
11556, 114syl5bi 232 . . . . . . . . . . . . . 14 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
116115rexlimdv 3023 . . . . . . . . . . . . 13 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin)𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
11755, 116syld 47 . . . . . . . . . . . 12 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
1181173exp 1261 . . . . . . . . . . 11 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑐𝐽 → (𝑋 = 𝑐 → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
119118com34 91 . . . . . . . . . 10 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑐𝐽 → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
120119com23 86 . . . . . . . . 9 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝑐𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
1214, 120syl7bi 245 . . . . . . . 8 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
122121ralrimdv 2962 . . . . . . 7 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
123 fibas 20692 . . . . . . . . 9 (fi‘𝑥) ∈ TopBases
124 tgcl 20684 . . . . . . . . 9 ((fi‘𝑥) ∈ TopBases → (topGen‘(fi‘𝑥)) ∈ Top)
125123, 124ax-mp 5 . . . . . . . 8 (topGen‘(fi‘𝑥)) ∈ Top
126 eleq1 2686 . . . . . . . 8 (𝐽 = (topGen‘(fi‘𝑥)) → (𝐽 ∈ Top ↔ (topGen‘(fi‘𝑥)) ∈ Top))
127125, 126mpbiri 248 . . . . . . 7 (𝐽 = (topGen‘(fi‘𝑥)) → 𝐽 ∈ Top)
128122, 127jctild 565 . . . . . 6 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
1291iscmp 21101 . . . . . 6 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
130128, 129syl6ibr 242 . . . . 5 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) → 𝐽 ∈ Comp))
1313, 130syld 47 . . . 4 (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) → 𝐽 ∈ Comp))
132131imp 445 . . 3 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) → 𝐽 ∈ Comp)
133132exlimiv 1855 . 2 (∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) → 𝐽 ∈ Comp)
1342, 133impbii 199 1 (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  {crab 2911  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   ciun 4485   class class class wbr 4613  ran crn 5075   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  cdom 7897  Fincfn 7899  ficfi 8260  topGenctg 16019  Topctop 20617  TopBasesctb 20620  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-ac2 9229
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-rpss 6890  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-card 8709  df-ac 8883  df-topgen 16025  df-top 20621  df-bases 20622  df-cmp 21100
This theorem is referenced by: (None)
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