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Theorem ptcmplem5 23560
Description: Lemma for ptcmp 23562. (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 4199 . 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 23556 . . 3 (πœ‘ β†’ (𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}) ∧ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋})))))
87simpld 496 . 2 (πœ‘ β†’ 𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}))
97simprd 497 . 2 (πœ‘ β†’ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
10 elpwi 4610 . . . . . 6 (𝑦 ∈ 𝒫 ran 𝑆 β†’ 𝑦 βŠ† ran 𝑆)
115ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝐴 ∈ 𝑉)
126ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝐹:𝐴⟢Comp)
131ad2antrr 725 . . . . . . . . 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 776 . . . . . . . . 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 777 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧) β†’ 𝑋 = βˆͺ 𝑦)
16 simpr 486 . . . . . . . . 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 6056 . . . . . . . . . . 11 (𝑧 = 𝑒 β†’ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑧) = (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1817eleq1d 2819 . . . . . . . . . 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 3443 . . . . . . . . 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 23559 . . . . . . . 8 Β¬ ((πœ‘ ∧ (𝑦 βŠ† ran 𝑆 ∧ 𝑋 = βˆͺ 𝑦)) ∧ Β¬ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
21 iman 403 . . . . . . . 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 458 . . . . . 6 ((πœ‘ ∧ 𝑦 βŠ† ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
2410, 23sylan2 594 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝒫 ran 𝑆) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
2524adantlr 714 . . . 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 4608 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 βˆͺ {𝑋}) ↔ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}))
27 eldif 3959 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 βˆͺ {𝑋}) βˆ– 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 βˆͺ {𝑋}) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆))
28 elpwunsn 4688 . . . . . . . 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 583 . . . . . 6 ((𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
3130adantll 713 . . . . 5 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ Β¬ 𝑦 ∈ 𝒫 ran 𝑆) β†’ 𝑋 ∈ 𝑦)
32 snssi 4812 . . . . . . . . 9 (𝑋 ∈ 𝑦 β†’ {𝑋} βŠ† 𝑦)
3332adantl 483 . . . . . . . 8 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ {𝑋} βŠ† 𝑦)
34 snfi 9044 . . . . . . . 8 {𝑋} ∈ Fin
35 elfpw 9354 . . . . . . . 8 ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} βŠ† 𝑦 ∧ {𝑋} ∈ Fin))
3633, 34, 35sylanblrc 591 . . . . . . 7 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37 unisng 4930 . . . . . . . . 9 (𝑋 ∈ 𝑦 β†’ βˆͺ {𝑋} = 𝑋)
3837eqcomd 2739 . . . . . . . 8 (𝑋 ∈ 𝑦 β†’ 𝑋 = βˆͺ {𝑋})
3938adantl 483 . . . . . . 7 (((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) ∧ 𝑋 ∈ 𝑦) β†’ 𝑋 = βˆͺ {𝑋})
40 unieq 4920 . . . . . . . 8 (𝑧 = {𝑋} β†’ βˆͺ 𝑧 = βˆͺ {𝑋})
4140rspceeqv 3634 . . . . . . 7 (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = βˆͺ {𝑋}) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
4236, 39, 41syl2anc 585 . . . . . 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 592 . . . 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 812 . . 3 ((πœ‘ ∧ 𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋})) β†’ (𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧))
4645impr 456 . 2 ((πœ‘ ∧ (𝑦 βŠ† (ran 𝑆 βˆͺ {𝑋}) ∧ 𝑋 = βˆͺ 𝑦)) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)𝑋 = βˆͺ 𝑧)
472, 8, 9, 46alexsub 23549 1 (πœ‘ β†’ (∏tβ€˜πΉ) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544   ∈ cmpo 7411  Xcixp 8891  Fincfn 8939  ficfi 9405  cardccrd 9930  topGenctg 17383  βˆtcpt 17384  Compccmp 22890  UFLcufl 23404
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-wdom 9560  df-card 9934  df-acn 9937  df-topgen 17389  df-pt 17390  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-cmp 22891  df-fil 23350  df-ufil 23405  df-ufl 23406  df-flim 23443  df-fcls 23445
This theorem is referenced by:  ptcmpg  23561
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