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Theorem cvmfolem 32410
Description: Lemma for cvmfo 32431. (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 32393 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2 cvmseu.1 . . . 4 𝐵 = 𝐶
3 cvmfolem.2 . . . 4 𝑋 = 𝐽
42, 3cnf 21770 . . 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 32394 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝑋) → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅))
87ex 413 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅)))
9 n0 4314 . . . . . . 7 ((𝑆𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆𝑧))
106cvmsn0 32399 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆𝑧) → 𝑤 ≠ ∅)
1110ad2antll 725 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → 𝑤 ≠ ∅)
12 n0 4314 . . . . . . . . . . 11 (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡𝑤)
1311, 12sylib 219 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → ∃𝑡 𝑡𝑤)
14 simprlr 776 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤 ∈ (𝑆𝑧))
156cvmsss 32398 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑆𝑧) → 𝑤𝐶)
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤𝐶)
17 simprr 769 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝑤)
1816, 17sseldd 3972 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐶)
19 elssuni 4866 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
2018, 19syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡 𝐶)
2120, 2sseqtrrdi 4022 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐵)
22 simpll 763 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
236cvmsf1o 32403 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆𝑧) ∧ 𝑡𝑤) → (𝐹𝑡):𝑡1-1-onto𝑧)
2422, 14, 17, 23syl3anc 1365 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑡1-1-onto𝑧)
25 f1ocnv 6624 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑡1-1-onto𝑧(𝐹𝑡):𝑧1-1-onto𝑡)
26 f1of 6612 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑧1-1-onto𝑡(𝐹𝑡):𝑧𝑡)
2724, 25, 263syl 18 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑧𝑡)
28 simprll 775 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥𝑧)
2927, 28ffvelrnd 6848 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝑡)
3021, 29sseldd 3972 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝐵)
31 f1ocnvfv2 7028 . . . . . . . . . . . . . . 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 6686 . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥 = (𝐹‘((𝐹𝑡)‘𝑥)))
36 fveq2 6667 . . . . . . . . . . . . . 14 (𝑦 = ((𝐹𝑡)‘𝑥) → (𝐹𝑦) = (𝐹‘((𝐹𝑡)‘𝑥)))
3736rspceeqv 3642 . . . . . . . . . . . . 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 457 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → (𝑡𝑤 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4039exlimdv 1927 . . . . . . . . . 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 457 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4342exlimdv 1927 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → (∃𝑤 𝑤 ∈ (𝑆𝑧) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
449, 43syl5bi 243 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ 𝑥𝑧) → ((𝑆𝑧) ≠ ∅ → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4544expimpd 454 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4645rexlimdva 3289 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅) → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
478, 46syld 47 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
4847ralrimiv 3186 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑦𝐵 𝑥 = (𝐹𝑦))
49 dffo3 6864 . 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 396   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   cuni 4837  cmpt 5143  ccnv 5553  cres 5556  cima 5557  wf 6348  ontowfo 6350  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  t crest 16684   Cn ccn 21748  Homeochmeo 22277   CovMap ccvm 32386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-fin 8502  df-fi 8864  df-rest 16686  df-topgen 16707  df-top 21418  df-topon 21435  df-bases 21470  df-cn 21751  df-hmeo 22279  df-cvm 32387
This theorem is referenced by:  cvmfo  32431
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