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Theorem ptcmplem1 23426
Description: Lemma for ptcmp 23432. (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 6673 . . . . . . 7 (πœ‘ β†’ 𝐹 Fn 𝐴)
4 eqid 2733 . . . . . . . 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 22944 . . . . . . 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 585 . . . . . 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 22769 . . . . . . . . . . 11 (π‘₯ ∈ Comp β†’ π‘₯ ∈ Top)
87ssriv 3952 . . . . . . . . . 10 Comp βŠ† Top
9 fss 6689 . . . . . . . . . 10 ((𝐹:𝐴⟢Comp ∧ Comp βŠ† Top) β†’ 𝐹:𝐴⟢Top)
102, 8, 9sylancl 587 . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝐴⟢Top)
11 ptcmp.2 . . . . . . . . . 10 𝑋 = X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)
124, 11ptbasfi 22955 . . . . . . . . 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 585 . . . . . . . 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 4117 . . . . . . . . . 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 5896 . . . . . . . . . . 11 ran 𝑆 = ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1716uneq2i 4124 . . . . . . . . . 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 2761 . . . . . . . . 9 (ran 𝑆 βˆͺ {𝑋}) = ({𝑋} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))
1918fveq2i 6849 . . . . . . . 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 2791 . . . . . . 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 6850 . . . . . 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 2773 . . . . 5 (πœ‘ β†’ (∏tβ€˜πΉ) = (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
2322unieqd 4883 . . . 4 (πœ‘ β†’ βˆͺ (∏tβ€˜πΉ) = βˆͺ (topGenβ€˜(fiβ€˜(ran 𝑆 βˆͺ {𝑋}))))
24 fibas 22350 . . . . 5 (fiβ€˜(ran 𝑆 βˆͺ {𝑋})) ∈ TopBases
25 unitg 22340 . . . . 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 2789 . . 3 (πœ‘ β†’ βˆͺ (∏tβ€˜πΉ) = βˆͺ (fiβ€˜(ran 𝑆 βˆͺ {𝑋})))
28 eqid 2733 . . . . . 6 (∏tβ€˜πΉ) = (∏tβ€˜πΉ)
2928ptuni 22968 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = βˆͺ (∏tβ€˜πΉ))
301, 10, 29syl2anc 585 . . . 4 (πœ‘ β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = βˆͺ (∏tβ€˜πΉ))
3111, 30eqtrid 2785 . . 3 (πœ‘ β†’ 𝑋 = βˆͺ (∏tβ€˜πΉ))
32 ptcmp.5 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5338 . . . . . 6 (πœ‘ β†’ 𝒫 𝑋 ∈ V)
34 eqid 2733 . . . . . . . . . . . 12 (𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) = (𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜))
3534mptpreima 6194 . . . . . . . . . . 11 (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) = {𝑀 ∈ 𝑋 ∣ (π‘€β€˜π‘˜) ∈ 𝑒}
3635ssrab3 4044 . . . . . . . . . 10 (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) βŠ† 𝑋
3732adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜))) β†’ 𝑋 ∈ (UFL ∩ dom card))
38 elpw2g 5305 . . . . . . . . . . 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 258 . . . . . . . . 9 ((πœ‘ ∧ (π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜))) β†’ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋)
4140ralrimivva 3194 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐴 βˆ€π‘’ ∈ (πΉβ€˜π‘˜)(β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) ∈ 𝒫 𝑋)
4215fmpox 8003 . . . . . . . 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 6680 . . . . . 6 (πœ‘ β†’ ran 𝑆 βŠ† 𝒫 𝑋)
4533, 44ssexd 5285 . . . . 5 (πœ‘ β†’ ran 𝑆 ∈ V)
46 snex 5392 . . . . 5 {𝑋} ∈ V
47 unexg 7687 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) β†’ (ran 𝑆 βˆͺ {𝑋}) ∈ V)
4845, 46, 47sylancl 587 . . . 4 (πœ‘ β†’ (ran 𝑆 βˆͺ {𝑋}) ∈ V)
49 fiuni 9372 . . . 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 2783 . 2 (πœ‘ β†’ 𝑋 = βˆͺ (ran 𝑆 βˆͺ {𝑋}))
5251, 22jca 513 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   βˆ– cdif 3911   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  βˆͺ ciun 4958   ↦ cmpt 5192   Γ— cxp 5635  β—‘ccnv 5636  dom cdm 5637  ran crn 5638   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500   ∈ cmpo 7363  Xcixp 8841  Fincfn 8889  ficfi 9354  cardccrd 9879  topGenctg 17327  βˆtcpt 17328  Topctop 22265  TopBasesctb 22318  Compccmp 22760  UFLcufl 23274
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-ixp 8842  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-topgen 17333  df-pt 17334  df-top 22266  df-bases 22319  df-cmp 22761
This theorem is referenced by:  ptcmplem5  23430
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