Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmfolem Structured version   Visualization version   GIF version

Theorem cvmfolem 35264
Description: Lemma for cvmfo 35285. (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 35247 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2 cvmseu.1 . . . 4 𝐵 = 𝐶
3 cvmfolem.2 . . . 4 𝑋 = 𝐽
42, 3cnf 23270 . . 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 35248 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝑋) → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅))
87ex 412 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥𝑋 → ∃𝑧𝐽 (𝑥𝑧 ∧ (𝑆𝑧) ≠ ∅)))
9 n0 4359 . . . . . . 7 ((𝑆𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆𝑧))
106cvmsn0 35253 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆𝑧) → 𝑤 ≠ ∅)
1110ad2antll 729 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → 𝑤 ≠ ∅)
12 n0 4359 . . . . . . . . . . 11 (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡𝑤)
1311, 12sylib 218 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ (𝑥𝑧𝑤 ∈ (𝑆𝑧))) → ∃𝑡 𝑡𝑤)
14 simprlr 780 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤 ∈ (𝑆𝑧))
156cvmsss 35252 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑆𝑧) → 𝑤𝐶)
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑤𝐶)
17 simprr 773 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝑤)
1816, 17sseldd 3996 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐶)
19 elssuni 4942 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
2018, 19syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡 𝐶)
2120, 2sseqtrrdi 4047 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑡𝐵)
22 simpll 767 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
236cvmsf1o 35257 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆𝑧) ∧ 𝑡𝑤) → (𝐹𝑡):𝑡1-1-onto𝑧)
2422, 14, 17, 23syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑡1-1-onto𝑧)
25 f1ocnv 6861 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑡1-1-onto𝑧(𝐹𝑡):𝑧1-1-onto𝑡)
26 f1of 6849 . . . . . . . . . . . . . . . 16 ((𝐹𝑡):𝑧1-1-onto𝑡(𝐹𝑡):𝑧𝑡)
2724, 25, 263syl 18 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → (𝐹𝑡):𝑧𝑡)
28 simprll 779 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥𝑧)
2927, 28ffvelcdmd 7105 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝑡)
3021, 29sseldd 3996 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → ((𝐹𝑡)‘𝑥) ∈ 𝐵)
31 f1ocnvfv2 7297 . . . . . . . . . . . . . . 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 6926 . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧𝐽) ∧ ((𝑥𝑧𝑤 ∈ (𝑆𝑧)) ∧ 𝑡𝑤)) → 𝑥 = (𝐹‘((𝐹𝑡)‘𝑥)))
36 fveq2 6907 . . . . . . . . . . . . . 14 (𝑦 = ((𝐹𝑡)‘𝑥) → (𝐹𝑦) = (𝐹‘((𝐹𝑡)‘𝑥)))
3736rspceeqv 3645 . . . . . . . . . . . . 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 1931 . . . . . . . . . 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 1931 . . . . . . 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 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 7122 . 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 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  cres 5691  cima 5692  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  t crest 17467   Cn ccn 23248  Homeochmeo 23777   CovMap ccvm 35240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-hmeo 23779  df-cvm 35241
This theorem is referenced by:  cvmfo  35285
  Copyright terms: Public domain W3C validator