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Theorem cvmfolem 34825
Description: Lemma for cvmfo 34846. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypotheses
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmseu.1 𝐵 = 𝐶
cvmfolem.2 𝑋 = 𝐽
Assertion
Ref Expression
cvmfolem (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)
Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑣,𝐵
Allowed substitution hints:   𝐵(𝑢,𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑋(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmfolem
Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmcn 34808 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2 cvmseu.1 . . . 4 𝐵 = 𝐶
3 cvmfolem.2 . . . 4 𝑋 = 𝐽
42, 3cnf 23137 . . 3 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
51, 4syl 17 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵𝑋)
6 cvmcov.1 . . . . . 6 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
76, 3cvmcov 34809 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝑋) → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅))
87ex 412 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅)))
9 n0 4342 . . . . . . 7 ((𝑆𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆𝑧))
106cvmsn0 34814 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆𝑧) → 𝑤 ≠ ∅)
1110ad2antll 728 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → 𝑤 ≠ ∅)
12 n0 4342 . . . . . . . . . . 11 (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡𝑤)
1311, 12sylib 217 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → ∃𝑡 𝑡𝑤)
14 simprlr 779 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤 ∈ (𝑆𝑧))
156cvmsss 34813 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑆𝑧) → 𝑤𝐶)
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤𝐶)
17 simprr 772 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝑤)
1816, 17sseldd 3979 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐶)
19 elssuni 4935 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
2018, 19syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡 𝐶)
2120, 2sseqtrrdi 4029 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐵)
22 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
236cvmsf1o 34818 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆𝑧) ∧ 𝑡𝑤) → (𝐹𝑡):𝑡1-1-onto𝑧)
2422, 14, 17, 23syl3anc 1369 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑡1-1-onto𝑧)
25 f1ocnv 6845 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑡1-1-onto𝑧(𝐹𝑡):𝑧1-1-onto𝑡)
26 f1of 6833 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑧1-1-onto𝑡(𝐹𝑡):𝑧𝑡)
2724, 25, 263syl 18 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑧𝑡)
28 simprll 778 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥𝑧)
2927, 28ffvelcdmd 7089 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝑡)
3021, 29sseldd 3979 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝐵)
31 f1ocnvfv2 7280 . . . . . . . . . . . . . . 15 (((𝐹𝑡):𝑡1-1-onto𝑧𝑥𝑧) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
3224, 28, 31syl2anc 583 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
33 fvres 6910 . . . . . . . . . . . . . . 15 (((𝐹𝑡)‘𝑥) ∈ 𝑡 → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = (𝐹‘((𝐹𝑡)‘𝑥)))
3429, 33syl 17 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = (𝐹‘((𝐹𝑡)‘𝑥)))
3532, 34eqtr3d 2769 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥 = (𝐹‘((𝐹𝑡)‘𝑥)))
36 fveq2 6891 . . . . . . . . . . . . . 14 (𝑦 = ((𝐹𝑡)‘𝑥) → (𝐹𝑦) = (𝐹‘((𝐹𝑡)‘𝑥)))
3736rspceeqv 3629 . . . . . . . . . . . . 13 ((((𝐹𝑡)‘𝑥) ∈ 𝐵𝑥 = (𝐹‘((𝐹𝑡)‘𝑥))) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
3830, 35, 37syl2anc 583 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
3938expr 456 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → (𝑡𝑤 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4039exlimdv 1929 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → (∃𝑡 𝑡𝑤 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4113, 40mpd 15 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
4241expr 456 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4342exlimdv 1929 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (∃𝑤 𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
449, 43biimtrid 241 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → ((𝑆𝑧) ≠ ∅ → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4544expimpd 453 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4645rexlimdva 3150 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
478, 46syld 47 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4847ralrimiv 3140 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦))
49 dffo3 7106 . 2 (𝐹:𝐵onto𝑋 ↔ (𝐹:𝐵𝑋 ∧ ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦)))
505, 48, 49sylanbrc 582 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  wne 2935  wral 3056  wrex 3065  {crab 3427  cdif 3941  cin 3943  wss 3944  c0 4318  𝒫 cpw 4598  {csn 4624   cuni 4903  cmpt 5225  ccnv 5671  cres 5674  cima 5675  wf 6538  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  t crest 17393   Cn ccn 23115  Homeochmeo 23644   CovMap ccvm 34801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-map 8838  df-en 8956  df-fin 8959  df-fi 9426  df-rest 17395  df-topgen 17416  df-top 22783  df-topon 22800  df-bases 22836  df-cn 23118  df-hmeo 23646  df-cvm 34802
This theorem is referenced by:  cvmfo  34846
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