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Theorem ptcmplem5 23780
Description: Lemma for ptcmp 23782. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
ptcmp.1 𝑆 = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
ptcmp.2 𝑋 = X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)
ptcmp.3 (πœ‘ β†’ 𝐴 ∈ 𝑉)
ptcmp.4 (πœ‘ β†’ 𝐹:𝐴⟢Comp)
ptcmp.5 (πœ‘ β†’ 𝑋 ∈ (UFL ∩ dom card))
Assertion
Ref Expression
ptcmplem5 (πœ‘ β†’ (∏tβ€˜πΉ) ∈ Comp)
Distinct variable groups:   π‘˜,𝑛,𝑒,𝑀,𝐴   𝑆,π‘˜,𝑛,𝑒   πœ‘,π‘˜,𝑛,𝑒   π‘˜,𝑉,𝑛,𝑒,𝑀   π‘˜,𝐹,𝑛,𝑒,𝑀   π‘˜,𝑋,𝑛,𝑒,𝑀
Allowed substitution hints:   πœ‘(𝑀)   𝑆(𝑀)
Proof of Theorem ptcmplem5
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.5 . . 3 (πœ‘ β†’ 𝑋 ∈ (UFL ∩ dom card))
21elin1d 4197 . 2 (πœ‘ β†’ 𝑋 ∈ UFL)
3 ptcmp.1 . . . 4 𝑆 = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
4 ptcmp.2 . . . 4 𝑋 = X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)
5 ptcmp.3 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑉)
6 ptcmp.4 . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢Comp)
73, 4, 5, 6, 1ptcmplem1 23776 . . 3 (πœ‘ β†’ (𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}) ∧ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋})))))
87simpld 493 . 2 (πœ‘ β†’ 𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}))
97simprd 494 . 2 (πœ‘ β†’ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
10 elpwi 4608 . . . . . 6 (𝑦 ∈ 𝒫 ran 𝑆 β†’ 𝑦 βŠ† ran 𝑆)
115ad2antrr 722 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝐴 ∈ 𝑉)
126ad2antrr 722 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝐹:𝐴⟢Comp)
131ad2antrr 722 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝑋 ∈ (UFL ∩ dom card))
14 simplrl 773 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝑦 βŠ† ran 𝑆)
15 simplrr 774 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝑋 = βˆͺ 𝑦)
16 simpr 483 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
17 imaeq2 6054 . . . . . . . . . . 11 (𝑧 = 𝑒 β†’ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑧) = (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1817eleq1d 2816 . . . . . . . . . 10 (𝑧 = 𝑒 β†’ ((β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑧) ∈ 𝑦 ↔ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝑦))
1918cbvrabv 3440 . . . . . . . . 9 {𝑧 ∈ (πΉβ€˜π‘˜) ∣ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑧) ∈ 𝑦} = {𝑒 ∈ (πΉβ€˜π‘˜) ∣ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝑦}
203, 4, 11, 12, 13, 14, 15, 16, 19ptcmplem4 23779 . . . . . . . 8 Β¬ ((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
21 iman 400 . . . . . . . 8 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) ↔ Β¬ ((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
2220, 21mpbir 230 . . . . . . 7 ((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
2322expr 455 . . . . . 6 ((πœ‘ ∧ 𝑦 βŠ† ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
2410, 23sylan2 591 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝒫 ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
2524adantlr 711 . . . 4 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
26 velpw 4606 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 βˆͺ {𝑋}) ↔ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}))
27 eldif 3957 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 βˆͺ {𝑋}) βˆ– 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 βˆͺ {𝑋}) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆))
28 elpwunsn 4686 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 βˆͺ {𝑋}) βˆ– 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
2927, 28sylbir 234 . . . . . . 7 ((𝑦 ∈ 𝒫 (ran 𝑆 βˆͺ {𝑋}) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
3026, 29sylanbr 580 . . . . . 6 ((𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
3130adantll 710 . . . . 5 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
32 snssi 4810 . . . . . . . . 9 (𝑋 ∈ 𝑦 β†’ {𝑋} βŠ† 𝑦)
3332adantl 480 . . . . . . . 8 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ {𝑋} βŠ† 𝑦)
34 snfi 9046 . . . . . . . 8 {𝑋} ∈ Fin
35 elfpw 9356 . . . . . . . 8 ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} βŠ† 𝑦 ∧ {𝑋} ∈ Fin))
3633, 34, 35sylanblrc 588 . . . . . . 7 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37 unisng 4928 . . . . . . . . 9 (𝑋 ∈ 𝑦 β†’ βˆͺ {𝑋} = 𝑋)
3837eqcomd 2736 . . . . . . . 8 (𝑋 ∈ 𝑦 β†’ 𝑋 = βˆͺ {𝑋})
3938adantl 480 . . . . . . 7 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ 𝑋 = βˆͺ {𝑋})
40 unieq 4918 . . . . . . . 8 (𝑧 = {𝑋} β†’ βˆͺ 𝑧 = βˆͺ {𝑋})
4140rspceeqv 3632 . . . . . . 7 (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = βˆͺ {𝑋}) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
4236, 39, 41syl2anc 582 . . . . . 6 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
4342a1d 25 . . . . 5 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
4431, 43syldan 589 . . . 4 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
4525, 44pm2.61dan 809 . . 3 ((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
4645impr 453 . 2 ((πœ‘ ∧ (𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}) ∧ 𝑋 = βˆͺ 𝑦)) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
472, 8, 9, 46alexsub 23769 1 (πœ‘ β†’ (∏tβ€˜πΉ) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  {crab 3430   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   ↦ cmpt 5230  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β€œ cima 5678  βŸΆwf 6538  β€˜cfv 6542   ∈ cmpo 7413  Xcixp 8893  Fincfn 8941  ficfi 9407  cardccrd 9932  topGenctg 17387  βˆtcpt 17388  Compccmp 23110  UFLcufl 23624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  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-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-wdom 9562  df-card 9936  df-acn 9939  df-topgen 17393  df-pt 17394  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cmp 23111  df-fil 23570  df-ufil 23625  df-ufl 23626  df-flim 23663  df-fcls 23665
This theorem is referenced by:  ptcmpg  23781
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