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Theorem pwfilemOLD 9342
Description: Obsolete version of pwfilem 9173 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 ∪ {𝑥}))
Assertion
Ref Expression
pwfilemOLD (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Distinct variable groups:   𝑏,𝑐   𝑥,𝑐
Allowed substitution hints:   𝐹(𝑥,𝑏,𝑐)

Proof of Theorem pwfilemOLD
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 pwundif 4625 . 2 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2 vex 3479 . . . . . . . . 9 𝑐 ∈ V
3 snex 5430 . . . . . . . . 9 {𝑥} ∈ V
42, 3unex 7728 . . . . . . . 8 (𝑐 ∪ {𝑥}) ∈ V
5 pwfilemOLD.1 . . . . . . . 8 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
64, 5fnmpti 6690 . . . . . . 7 𝐹 Fn 𝒫 𝑏
7 dffn4 6808 . . . . . . 7 (𝐹 Fn 𝒫 𝑏𝐹:𝒫 𝑏onto→ran 𝐹)
86, 7mpbi 229 . . . . . 6 𝐹:𝒫 𝑏onto→ran 𝐹
9 fodomfi 9321 . . . . . 6 ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
108, 9mpan2 690 . . . . 5 (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11 domfi 9188 . . . . 5 ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
1210, 11mpdan 686 . . . 4 (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13 eldifi 4125 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
143elpwun 7751 . . . . . . . . 9 (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
1513, 14sylib 217 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16 undif1 4474 . . . . . . . . 9 ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17 elpwunsn 4686 . . . . . . . . . . 11 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥𝑑)
1817snssd 4811 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19 ssequn2 4182 . . . . . . . . . 10 ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
2018, 19sylib 217 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
2116, 20eqtr2id 2786 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22 uneq1 4155 . . . . . . . . 9 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
2322rspceeqv 3632 . . . . . . . 8 (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2415, 21, 23syl2anc 585 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
255, 4elrnmpti 5957 . . . . . . 7 (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2624, 25sylibr 233 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
2726ssriv 3985 . . . . 5 (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28 ssdomg 8992 . . . . 5 (ran 𝐹 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹 → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹))
2912, 27, 28mpisyl 21 . . . 4 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹)
30 domfi 9188 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
3112, 29, 30syl2anc 585 . . 3 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
32 unfi 9168 . . 3 (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
3331, 32mpancom 687 . 2 (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
341, 33eqeltrid 2838 1 (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3071  cdif 3944  cun 3945  wss 3947  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  cmpt 5230  ran crn 5676   Fn wfn 6535  ontowfo 6538  cdom 8933  Fincfn 8935
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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  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-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-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-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7851  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-fin 8939
This theorem is referenced by: (None)
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