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Theorem cvmfolem 35285
Description: Lemma for cvmfo 35306. (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 35268 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2 cvmseu.1 . . . 4 𝐵 = 𝐶
3 cvmfolem.2 . . . 4 𝑋 = 𝐽
42, 3cnf 23255 . . 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 35269 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝑋) → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅))
87ex 412 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅)))
9 n0 4352 . . . . . . 7 ((𝑆𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆𝑧))
106cvmsn0 35274 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆𝑧) → 𝑤 ≠ ∅)
1110ad2antll 729 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → 𝑤 ≠ ∅)
12 n0 4352 . . . . . . . . . . 11 (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡𝑤)
1311, 12sylib 218 . . . . . . . . . 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 35273 . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐶)
19 elssuni 4936 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
2018, 19syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡 𝐶)
2120, 2sseqtrrdi 4024 . . . . . . . . . . . . . 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 35278 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆𝑧) ∧ 𝑡𝑤) → (𝐹𝑡):𝑡1-1-onto𝑧)
2422, 14, 17, 23syl3anc 1372 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑡1-1-onto𝑧)
25 f1ocnv 6859 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑡1-1-onto𝑧(𝐹𝑡):𝑧1-1-onto𝑡)
26 f1of 6847 . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝑡)
3021, 29sseldd 3983 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝐵)
31 f1ocnvfv2 7298 . . . . . . . . . . . . . . 15 (((𝐹𝑡):𝑡1-1-onto𝑧𝑥𝑧) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
3224, 28, 31syl2anc 584 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
33 fvres 6924 . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥 = (𝐹‘((𝐹𝑡)‘𝑥)))
36 fveq2 6905 . . . . . . . . . . . . . 14 (𝑦 = ((𝐹𝑡)‘𝑥) → (𝐹𝑦) = (𝐹‘((𝐹𝑡)‘𝑥)))
3736rspceeqv 3644 . . . . . . . . . . . . 13 ((((𝐹𝑡)‘𝑥) ∈ 𝐵𝑥 = (𝐹‘((𝐹𝑡)‘𝑥))) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
3830, 35, 37syl2anc 584 . . . . . . . . . . . 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 1932 . . . . . . . . . 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 1932 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (∃𝑤 𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
449, 43biimtrid 242 . . . . . 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 3154 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
478, 46syld 47 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4847ralrimiv 3144 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦))
49 dffo3 7121 . 2 (𝐹:𝐵onto𝑋 ↔ (𝐹:𝐵𝑋 ∧ ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦)))
505, 48, 49sylanbrc 583 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  cdif 3947  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906  cmpt 5224  ccnv 5683  cres 5686  cima 5687  wf 6556  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  t crest 17466   Cn ccn 23233  Homeochmeo 23762   CovMap ccvm 35261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-map 8869  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cn 23236  df-hmeo 23764  df-cvm 35262
This theorem is referenced by:  cvmfo  35306
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