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Theorem pwfilemOLD 9113
Description: Obsolete version of pwfilem 8960 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 4559 . 2 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2 vex 3436 . . . . . . . . 9 𝑐 ∈ V
3 snex 5354 . . . . . . . . 9 {𝑥} ∈ V
42, 3unex 7596 . . . . . . . 8 (𝑐 ∪ {𝑥}) ∈ V
5 pwfilemOLD.1 . . . . . . . 8 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
64, 5fnmpti 6576 . . . . . . 7 𝐹 Fn 𝒫 𝑏
7 dffn4 6694 . . . . . . 7 (𝐹 Fn 𝒫 𝑏𝐹:𝒫 𝑏onto→ran 𝐹)
86, 7mpbi 229 . . . . . 6 𝐹:𝒫 𝑏onto→ran 𝐹
9 fodomfi 9092 . . . . . 6 ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
108, 9mpan2 688 . . . . 5 (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11 domfi 8975 . . . . 5 ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
1210, 11mpdan 684 . . . 4 (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13 eldifi 4061 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
143elpwun 7619 . . . . . . . . 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 4409 . . . . . . . . 9 ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17 elpwunsn 4619 . . . . . . . . . . 11 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥𝑑)
1817snssd 4742 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19 ssequn2 4117 . . . . . . . . . 10 ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
2018, 19sylib 217 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
2116, 20eqtr2id 2791 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22 uneq1 4090 . . . . . . . . 9 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
2322rspceeqv 3575 . . . . . . . 8 (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2415, 21, 23syl2anc 584 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
255, 4elrnmpti 5869 . . . . . . 7 (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2624, 25sylibr 233 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
2726ssriv 3925 . . . . 5 (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28 ssdomg 8786 . . . . 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 8975 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
3112, 29, 30syl2anc 584 . . 3 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
32 unfi 8955 . . 3 (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
3331, 32mpancom 685 . 2 (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
341, 33eqeltrid 2843 1 (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wrex 3065  cdif 3884  cun 3885  wss 3887  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  cmpt 5157  ran crn 5590   Fn wfn 6428  ontowfo 6431  cdom 8731  Fincfn 8733
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737
This theorem is referenced by: (None)
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