MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alexsubALT Structured version   Visualization version   GIF version

Theorem alexsubALT 22634
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 22630 . 2 (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
31alexsubALTlem4 22633 . . . . 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 velpw 4517 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
5 eleq2 2900 . . . . . . . . . . . . . . . . . . 19 (𝑋 = 𝑐 → (𝑡𝑋𝑡 𝑐))
653ad2ant3 1132 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋𝑡 𝑐))
7 eluni 4814 . . . . . . . . . . . . . . . . . . . 20 (𝑡 𝑐 ↔ ∃𝑤(𝑡𝑤𝑤𝑐))
8 ssel 3937 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐𝐽 → (𝑤𝑐𝑤𝐽))
9 eleq2 2900 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤𝐽𝑤 ∈ (topGen‘(fi‘𝑥))))
10 tg2 21548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑤 ∈ (topGen‘(fi‘𝑥)) ∧ 𝑡𝑤) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))
1110ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (topGen‘(fi‘𝑥)) → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤)))
129, 11syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤𝐽 → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))))
138, 12sylan9r 512 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑤𝑐 → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤))))
14133impia 1114 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (𝑡𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦𝑦𝑤)))
15 sseq2 3969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑤 → (𝑦𝑧𝑦𝑤))
1615rspcev 3600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑤𝑐𝑦𝑤) → ∃𝑧𝑐 𝑦𝑧)
1716ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤𝑐 → (𝑦𝑤 → ∃𝑧𝑐 𝑦𝑧))
18173ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → (𝑦𝑤 → ∃𝑧𝑐 𝑦𝑧))
1918anim2d 614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑤𝑐) → ((𝑡𝑦𝑦𝑤) → (𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
2019reximdv 3259 . . . . . . . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → ((𝑡𝑤𝑤𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
2524exlimdv 1935 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (∃𝑤(𝑡𝑤𝑤𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
267, 25syl5bi 245 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽) → (𝑡 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
27263adant3 1129 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
286, 27sylbid 243 . . . . . . . . . . . . . . . . 17 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋 → ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
29 ssel 3937 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑧 → (𝑡𝑦𝑡𝑧))
30 elunii 4816 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡𝑧𝑧𝑐) → 𝑡 𝑐)
3130expcom 417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑐 → (𝑡𝑧𝑡 𝑐))
326biimprd 251 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡 𝑐𝑡𝑋))
3331, 32sylan9r 512 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑧𝑐) → (𝑡𝑧𝑡𝑋))
3429, 33syl9r 78 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑧𝑐) → (𝑦𝑧 → (𝑡𝑦𝑡𝑋)))
3534rexlimdva 3270 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∃𝑧𝑐 𝑦𝑧 → (𝑡𝑦𝑡𝑋)))
3635com23 86 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑦 → (∃𝑧𝑐 𝑦𝑧𝑡𝑋)))
3736impd 414 . . . . . . . . . . . . . . . . . 18 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → ((𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧) → 𝑡𝑋))
3837rexlimdvw 3276 . . . . . . . . . . . . . . . . 17 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧) → 𝑡𝑋))
3928, 38impbid 215 . . . . . . . . . . . . . . . 16 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋 ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧)))
40 elunirab 4827 . . . . . . . . . . . . . . . 16 (𝑡 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡𝑦 ∧ ∃𝑧𝑐 𝑦𝑧))
4139, 40syl6bbr 292 . . . . . . . . . . . . . . 15 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑡𝑋𝑡 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
4241eqrdv 2819 . . . . . . . . . . . . . 14 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → 𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
43 ssrab2 4032 . . . . . . . . . . . . . . . 16 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ⊆ (fi‘𝑥)
44 fvex 6656 . . . . . . . . . . . . . . . . 17 (fi‘𝑥) ∈ V
4544elpw2 5221 . . . . . . . . . . . . . . . 16 ({𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∈ 𝒫 (fi‘𝑥) ↔ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ⊆ (fi‘𝑥))
4643, 45mpbir 234 . . . . . . . . . . . . . . 15 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∈ 𝒫 (fi‘𝑥)
47 unieq 4822 . . . . . . . . . . . . . . . . . 18 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → 𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
4847eqeq2d 2832 . . . . . . . . . . . . . . . . 17 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑋 = 𝑎𝑋 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
49 pweq 4528 . . . . . . . . . . . . . . . . . . 19 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → 𝒫 𝑎 = 𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧})
5049ineq1d 4163 . . . . . . . . . . . . . . . . . 18 (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝒫 𝑎 ∩ Fin) = (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin))
5150rexeqdv 3397 . . . . . . . . . . . . . . . . 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 348 . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . . 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 8802 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin) ↔ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∧ 𝑏 ∈ Fin))
57 ssel 3937 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑡𝑏𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧}))
58 sseq1 3968 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑡 → (𝑦𝑧𝑡𝑧))
5958rexbidv 3283 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑡 → (∃𝑧𝑐 𝑦𝑧 ↔ ∃𝑧𝑐 𝑡𝑧))
6059elrab 3657 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ↔ (𝑡 ∈ (fi‘𝑥) ∧ ∃𝑧𝑐 𝑡𝑧))
6160simprbi 500 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑧𝑐 𝑡𝑧)
6257, 61syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → (𝑡𝑏 → ∃𝑧𝑐 𝑡𝑧))
6362ralrimiv 3169 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∀𝑡𝑏𝑧𝑐 𝑡𝑧)
64 sseq2 3969 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝑓𝑡) → (𝑡𝑧𝑡 ⊆ (𝑓𝑡)))
6564ac6sfi 8738 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ Fin ∧ ∀𝑡𝑏𝑧𝑐 𝑡𝑧) → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)))
6665ex 416 . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . 19 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} → ∃𝑓(𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))))
69 simprll 778 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑓:𝑏𝑐)
70 frn 6493 . . . . . . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 ∈ Fin)
73 ffn 6487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:𝑏𝑐𝑓 Fn 𝑏)
74 dffn4 6569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 Fn 𝑏𝑓:𝑏onto→ran 𝑓)
7573, 74sylib 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:𝑏𝑐𝑓:𝑏onto→ran 𝑓)
7675adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) → 𝑓:𝑏onto→ran 𝑓)
7776ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑓:𝑏onto→ran 𝑓)
78 fodomfi 8773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏 ∈ Fin ∧ 𝑓:𝑏onto→ran 𝑓) → ran 𝑓𝑏)
7972, 77, 78syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝑏)
80 domfi 8715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ Fin ∧ ran 𝑓𝑏) → ran 𝑓 ∈ Fin)
8172, 79, 80syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓 ∈ Fin)
8271, 81jca 515 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → (ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin))
83 elin 3926 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ↔ (ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin))
84 vex 3474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐 ∈ V
8584elpw2 5221 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran 𝑓 ∈ 𝒫 𝑐 ↔ ran 𝑓𝑐)
8685anbi1i 626 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin) ↔ (ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin))
8783, 86bitr2i 279 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓𝑐 ∧ ran 𝑓 ∈ Fin) ↔ ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
8882, 87sylib 221 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
89 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 = 𝑏)
90 uniiun 4955 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑏 = 𝑡𝑏 𝑡
91 simprlr 779 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡))
92 ss2iun 4910 . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 93eqsstrid 3991 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 𝑡𝑏 (𝑓𝑡))
95 fniunfv 6980 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑏 ran 𝑓)
9889, 97eqsstrd 3981 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 ran 𝑓)
99 simpll2 1210 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑐𝐽)
10071, 99sstrd 3953 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ran 𝑓𝐽)
101 uniss 4819 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran 𝑓𝐽 ran 𝑓 𝐽)
102101, 1sseqtrrdi 3994 . . . . . . . . . . . . . . . . . . . . . . . 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 3960 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → 𝑋 = ran 𝑓)
105 unieq 4822 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = ran 𝑓 𝑑 = ran 𝑓)
106105eqeq2d 2832 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = ran 𝑓 → (𝑋 = 𝑑𝑋 = ran 𝑓))
107106rspcev 3600 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
10888, 104, 107syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) ∧ 𝑋 = 𝑏)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
109108exp32 424 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) ∧ 𝑏 ∈ Fin) → ((𝑓:𝑏𝑐 ∧ ∀𝑡𝑏 𝑡 ⊆ (𝑓𝑡)) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
110109exlimdv 1935 . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . 14 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐𝐽𝑋 = 𝑐) → (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧𝑐 𝑦𝑧} ∩ Fin) → (𝑋 = 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
116115rexlimdv 3269 . . . . . . . . . . . . 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 1116 . . . . . . . . . . 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 258 . . . . . . . 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 3176 . . . . . . 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 21560 . . . . . . . . 9 (fi‘𝑥) ∈ TopBases
124 tgcl 21552 . . . . . . . . 9 ((fi‘𝑥) ∈ TopBases → (topGen‘(fi‘𝑥)) ∈ Top)
125123, 124ax-mp 5 . . . . . . . 8 (topGen‘(fi‘𝑥)) ∈ Top
126 eleq1 2899 . . . . . . . 8 (𝐽 = (topGen‘(fi‘𝑥)) → (𝐽 ∈ Top ↔ (topGen‘(fi‘𝑥)) ∈ Top))
127125, 126mpbiri 261 . . . . . . 7 (𝐽 = (topGen‘(fi‘𝑥)) → 𝐽 ∈ Top)
128122, 127jctild 529 . . . . . 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 21971 . . . . . 6 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
130128, 129syl6ibr 255 . . . . 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 410 . . 3 ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) → 𝐽 ∈ Comp)
133132exlimiv 1932 . 2 (∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) → 𝐽 ∈ Comp)
1342, 133impbii 212 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 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wral 3126  wrex 3127  {crab 3130  cin 3909  wss 3910  𝒫 cpw 4512   cuni 4811   ciun 4892   class class class wbr 5039  ran crn 5529   Fn wfn 6323  wf 6324  ontowfo 6326  cfv 6328  cdom 8482  Fincfn 8484  ficfi 8850  topGenctg 16689  Topctop 21476  TopBasesctb 21528  Compccmp 21969
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-ac2 9862
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  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-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-rpss 7424  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-fin 8488  df-fi 8851  df-card 9344  df-ac 9519  df-topgen 16695  df-top 21477  df-bases 21529  df-cmp 21970
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator