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Theorem cvmfolem 33913
Description: Lemma for cvmfo 33934. (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 33896 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2 cvmseu.1 . . . 4 𝐵 = 𝐶
3 cvmfolem.2 . . . 4 𝑋 = 𝐽
42, 3cnf 22613 . . 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 33897 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝑋) → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅))
87ex 414 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅)))
9 n0 4311 . . . . . . 7 ((𝑆𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆𝑧))
106cvmsn0 33902 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆𝑧) → 𝑤 ≠ ∅)
1110ad2antll 728 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → 𝑤 ≠ ∅)
12 n0 4311 . . . . . . . . . . 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 33901 . . . . . . . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐶)
19 elssuni 4903 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
2018, 19syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡 𝐶)
2120, 2sseqtrrdi 4000 . . . . . . . . . . . . . 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 33906 . . . . . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑡1-1-onto𝑧(𝐹𝑡):𝑧1-1-onto𝑡)
26 f1of 6789 . . . . . . . . . . . . . . . 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 7041 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝑡)
3021, 29sseldd 3950 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝐵)
31 f1ocnvfv2 7228 . . . . . . . . . . . . . . 15 (((𝐹𝑡):𝑡1-1-onto𝑧𝑥𝑧) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
3224, 28, 31syl2anc 585 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘((𝐹𝑡)‘𝑥)) = 𝑥)
33 fvres 6866 . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥 = (𝐹‘((𝐹𝑡)‘𝑥)))
36 fveq2 6847 . . . . . . . . . . . . . 14 (𝑦 = ((𝐹𝑡)‘𝑥) → (𝐹𝑦) = (𝐹‘((𝐹𝑡)‘𝑥)))
3736rspceeqv 3600 . . . . . . . . . . . . 13 ((((𝐹𝑡)‘𝑥) ∈ 𝐵𝑥 = (𝐹‘((𝐹𝑡)‘𝑥))) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
3830, 35, 37syl2anc 585 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ∃𝑦𝐵 𝑥 = (𝐹𝑦))
3938expr 458 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → (𝑡𝑤 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4039exlimdv 1937 . . . . . . . . . 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 458 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4342exlimdv 1937 . . . . . . 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 455 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4645rexlimdva 3153 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
478, 46syld 47 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4847ralrimiv 3143 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦))
49 dffo3 7057 . 2 (𝐹:𝐵onto𝑋 ↔ (𝐹:𝐵𝑋 ∧ ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦)))
505, 48, 49sylanbrc 584 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  wrex 3074  {crab 3410  cdif 3912  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870  cmpt 5193  ccnv 5637  cres 5640  cima 5641  wf 6497  ontowfo 6499  1-1-ontowf1o 6500  cfv 6501  (class class class)co 7362  t crest 17309   Cn ccn 22591  Homeochmeo 23120   CovMap ccvm 33889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-map 8774  df-en 8891  df-fin 8894  df-fi 9354  df-rest 17311  df-topgen 17332  df-top 22259  df-topon 22276  df-bases 22312  df-cn 22594  df-hmeo 23122  df-cvm 33890
This theorem is referenced by:  cvmfo  33934
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