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Theorem pwfilemOLD 9349
Description: Obsolete version of pwfilem 9180 as of 7-Sep-2024. (Contributed by NM, 26-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
pwfilemOLD.1 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 βˆͺ {π‘₯}))
Assertion
Ref Expression
pwfilemOLD (𝒫 𝑏 ∈ Fin β†’ 𝒫 (𝑏 βˆͺ {π‘₯}) ∈ Fin)
Distinct variable groups:   𝑏,𝑐   π‘₯,𝑐
Allowed substitution hints:   𝐹(π‘₯,𝑏,𝑐)
Proof of Theorem pwfilemOLD
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 pwundif 4627 . 2 𝒫 (𝑏 βˆͺ {π‘₯}) = ((𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) βˆͺ 𝒫 𝑏)
2 vex 3477 . . . . . . . . 9 𝑐 ∈ V
3 snex 5432 . . . . . . . . 9 {π‘₯} ∈ V
42, 3unex 7736 . . . . . . . 8 (𝑐 βˆͺ {π‘₯}) ∈ V
5 pwfilemOLD.1 . . . . . . . 8 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 βˆͺ {π‘₯}))
64, 5fnmpti 6694 . . . . . . 7 𝐹 Fn 𝒫 𝑏
7 dffn4 6812 . . . . . . 7 (𝐹 Fn 𝒫 𝑏 ↔ 𝐹:𝒫 𝑏–ontoβ†’ran 𝐹)
86, 7mpbi 229 . . . . . 6 𝐹:𝒫 𝑏–ontoβ†’ran 𝐹
9 fodomfi 9328 . . . . . 6 ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏–ontoβ†’ran 𝐹) β†’ ran 𝐹 β‰Ό 𝒫 𝑏)
108, 9mpan2 688 . . . . 5 (𝒫 𝑏 ∈ Fin β†’ ran 𝐹 β‰Ό 𝒫 𝑏)
11 domfi 9195 . . . . 5 ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 β‰Ό 𝒫 𝑏) β†’ ran 𝐹 ∈ Fin)
1210, 11mpdan 684 . . . 4 (𝒫 𝑏 ∈ Fin β†’ ran 𝐹 ∈ Fin)
13 eldifi 4127 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ 𝑑 ∈ 𝒫 (𝑏 βˆͺ {π‘₯}))
143elpwun 7759 . . . . . . . . 9 (𝑑 ∈ 𝒫 (𝑏 βˆͺ {π‘₯}) ↔ (𝑑 βˆ– {π‘₯}) ∈ 𝒫 𝑏)
1513, 14sylib 217 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ (𝑑 βˆ– {π‘₯}) ∈ 𝒫 𝑏)
16 undif1 4476 . . . . . . . . 9 ((𝑑 βˆ– {π‘₯}) βˆͺ {π‘₯}) = (𝑑 βˆͺ {π‘₯})
17 elpwunsn 4688 . . . . . . . . . . 11 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ π‘₯ ∈ 𝑑)
1817snssd 4813 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ {π‘₯} βŠ† 𝑑)
19 ssequn2 4184 . . . . . . . . . 10 ({π‘₯} βŠ† 𝑑 ↔ (𝑑 βˆͺ {π‘₯}) = 𝑑)
2018, 19sylib 217 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ (𝑑 βˆͺ {π‘₯}) = 𝑑)
2116, 20eqtr2id 2784 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ 𝑑 = ((𝑑 βˆ– {π‘₯}) βˆͺ {π‘₯}))
22 uneq1 4157 . . . . . . . . 9 (𝑐 = (𝑑 βˆ– {π‘₯}) β†’ (𝑐 βˆͺ {π‘₯}) = ((𝑑 βˆ– {π‘₯}) βˆͺ {π‘₯}))
2322rspceeqv 3634 . . . . . . . 8 (((𝑑 βˆ– {π‘₯}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 βˆ– {π‘₯}) βˆͺ {π‘₯})) β†’ βˆƒπ‘ ∈ 𝒫 𝑏𝑑 = (𝑐 βˆͺ {π‘₯}))
2415, 21, 23syl2anc 583 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ βˆƒπ‘ ∈ 𝒫 𝑏𝑑 = (𝑐 βˆͺ {π‘₯}))
255, 4elrnmpti 5960 . . . . . . 7 (𝑑 ∈ ran 𝐹 ↔ βˆƒπ‘ ∈ 𝒫 𝑏𝑑 = (𝑐 βˆͺ {π‘₯}))
2624, 25sylibr 233 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β†’ 𝑑 ∈ ran 𝐹)
2726ssriv 3987 . . . . 5 (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) βŠ† ran 𝐹
28 ssdomg 8999 . . . . 5 (ran 𝐹 ∈ Fin β†’ ((𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) βŠ† ran 𝐹 β†’ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β‰Ό ran 𝐹))
2912, 27, 28mpisyl 21 . . . 4 (𝒫 𝑏 ∈ Fin β†’ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β‰Ό ran 𝐹)
30 domfi 9195 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) β‰Ό ran 𝐹) β†’ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) ∈ Fin)
3112, 29, 30syl2anc 583 . . 3 (𝒫 𝑏 ∈ Fin β†’ (𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) ∈ Fin)
32 unfi 9175 . . 3 (((𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) β†’ ((𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) βˆͺ 𝒫 𝑏) ∈ Fin)
3331, 32mpancom 685 . 2 (𝒫 𝑏 ∈ Fin β†’ ((𝒫 (𝑏 βˆͺ {π‘₯}) βˆ– 𝒫 𝑏) βˆͺ 𝒫 𝑏) ∈ Fin)
341, 33eqeltrid 2836 1 (𝒫 𝑏 ∈ Fin β†’ 𝒫 (𝑏 βˆͺ {π‘₯}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069   βˆ– cdif 3946   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678   Fn wfn 6539  β€“ontoβ†’wfo 6542   β‰Ό cdom 8940  Fincfn 8942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  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-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-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-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-om 7859  df-1o 8469  df-er 8706  df-en 8943  df-dom 8944  df-fin 8946
This theorem is referenced by: (None)
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