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Theorem ptcmplem1 23563
Description: Lemma for ptcmp 23569. (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
ptcmplem1 (πœ‘ β†’ (𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}) ∧ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋})))))
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 ptcmplem1
Dummy variables 𝑔 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.3 . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ 𝑉)
2 ptcmp.4 . . . . . . . 8 (πœ‘ β†’ 𝐹:𝐴⟢Comp)
32ffnd 6718 . . . . . . 7 (πœ‘ β†’ 𝐹 Fn 𝐴)
4 eqid 2732 . . . . . . . 8 {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))} = {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}
54ptval 23081 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ (∏tβ€˜πΉ) = (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}))
61, 3, 5syl2anc 584 . . . . . 6 (πœ‘ β†’ (∏tβ€˜πΉ) = (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}))
7 cmptop 22906 . . . . . . . . . . 11 (π‘₯ ∈ Comp β†’ π‘₯ ∈ Top)
87ssriv 3986 . . . . . . . . . 10 Comp βŠ† Top
9 fss 6734 . . . . . . . . . 10 ((𝐹:𝐴⟢Comp ∧ Comp βŠ† Top) β†’ 𝐹:𝐴⟢Top)
102, 8, 9sylancl 586 . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝐴⟢Top)
11 ptcmp.2 . . . . . . . . . 10 𝑋 = X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)
124, 11ptbasfi 23092 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))} = (fiβ€˜({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))))
131, 10, 12syl2anc 584 . . . . . . . 8 (πœ‘ β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))} = (fiβ€˜({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))))
14 uncom 4153 . . . . . . . . . 10 (ran 𝑆 βˆͺ {𝑋}) = ({𝑋} βˆͺ ran 𝑆)
15 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1615rneqi 5936 . . . . . . . . . . 11 ran 𝑆 = ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1716uneq2i 4160 . . . . . . . . . 10 ({𝑋} βˆͺ ran 𝑆) = ({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))
1814, 17eqtri 2760 . . . . . . . . 9 (ran 𝑆 βˆͺ {𝑋}) = ({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))
1918fveq2i 6894 . . . . . . . 8 (fiβ€˜(ran 𝑆 βˆͺ {𝑋})) = (fiβ€˜({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))))
2013, 19eqtr4di 2790 . . . . . . 7 (πœ‘ β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))} = (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
2120fveq2d 6895 . . . . . 6 (πœ‘ β†’ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
226, 21eqtrd 2772 . . . . 5 (πœ‘ β†’ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
2322unieqd 4922 . . . 4 (πœ‘ β†’ βˆͺ (∏tβ€˜πΉ) = βˆͺ (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
24 fibas 22487 . . . . 5 (fiβ€˜(ran 𝑆 βˆͺ {𝑋})) ∈ TopBases
25 unitg 22477 . . . . 5 ((fiβ€˜(ran 𝑆 βˆͺ {𝑋})) ∈ TopBases β†’ βˆͺ (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
2624, 25ax-mp 5 . . . 4 βˆͺ (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋}))
2723, 26eqtrdi 2788 . . 3 (πœ‘ β†’ βˆͺ (∏tβ€˜πΉ) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
28 eqid 2732 . . . . . 6 (∏tβ€˜πΉ) = (∏tβ€˜πΉ)
2928ptuni 23105 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = βˆͺ (∏tβ€˜πΉ))
301, 10, 29syl2anc 584 . . . 4 (πœ‘ β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = βˆͺ (∏tβ€˜πΉ))
3111, 30eqtrid 2784 . . 3 (πœ‘ β†’ 𝑋 = βˆͺ (∏tβ€˜πΉ))
32 ptcmp.5 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5377 . . . . . 6 (πœ‘ β†’ 𝒫 𝑋 ∈ V)
34 eqid 2732 . . . . . . . . . . . 12 (𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) = (𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜))
3534mptpreima 6237 . . . . . . . . . . 11 (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) = {𝑀 ∈ 𝑋 ∣ (π‘€β€˜π‘˜) ∈ 𝑒}
3635ssrab3 4080 . . . . . . . . . 10 (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) βŠ† 𝑋
3732adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜))) β†’ 𝑋 ∈ (UFL ∩ dom card))
38 elpw2g 5344 . . . . . . . . . . 11 (𝑋 ∈ (UFL ∩ dom card) β†’ ((β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋 ↔ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) βŠ† 𝑋))
3937, 38syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜))) β†’ ((β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋 ↔ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) βŠ† 𝑋))
4036, 39mpbiri 257 . . . . . . . . 9 ((πœ‘ ∧ (π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜))) β†’ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋)
4140ralrimivva 3200 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐴 βˆ€π‘’ ∈ (πΉβ€˜π‘˜)(β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋)
4215fmpox 8055 . . . . . . . 8 (βˆ€π‘˜ ∈ 𝐴 βˆ€π‘’ ∈ (πΉβ€˜π‘˜)(β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋 ↔ 𝑆:βˆͺ π‘˜ ∈ 𝐴 ({π‘˜} Γ— (πΉβ€˜π‘˜))βŸΆπ’« 𝑋)
4341, 42sylib 217 . . . . . . 7 (πœ‘ β†’ 𝑆:βˆͺ π‘˜ ∈ 𝐴 ({π‘˜} Γ— (πΉβ€˜π‘˜))βŸΆπ’« 𝑋)
4443frnd 6725 . . . . . 6 (πœ‘ β†’ ran 𝑆 βŠ† 𝒫 𝑋)
4533, 44ssexd 5324 . . . . 5 (πœ‘ β†’ ran 𝑆 ∈ V)
46 snex 5431 . . . . 5 {𝑋} ∈ V
47 unexg 7738 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) β†’ (ran 𝑆 βˆͺ {𝑋}) ∈ V)
4845, 46, 47sylancl 586 . . . 4 (πœ‘ β†’ (ran 𝑆 βˆͺ {𝑋}) ∈ V)
49 fiuni 9425 . . . 4 ((ran 𝑆 βˆͺ {𝑋}) ∈ V β†’ βˆͺ (ran 𝑆 βˆͺ {𝑋}) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
5048, 49syl 17 . . 3 (πœ‘ β†’ βˆͺ (ran 𝑆 βˆͺ {𝑋}) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
5127, 31, 503eqtr4d 2782 . 2 (πœ‘ β†’ 𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}))
5251, 22jca 512 1 (πœ‘ β†’ (𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}) ∧ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543   ∈ cmpo 7413  Xcixp 8893  Fincfn 8941  ficfi 9407  cardccrd 9932  topGenctg 17385  βˆtcpt 17386  Topctop 22402  TopBasesctb 22455  Compccmp 22897  UFLcufl 23411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-ixp 8894  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-topgen 17391  df-pt 17392  df-top 22403  df-bases 22456  df-cmp 22898
This theorem is referenced by:  ptcmplem5  23567
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