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Theorem pwfilem 8810
Description: Lemma for pwfi 8811. (Contributed by NM, 26-Mar-2007.)
Hypothesis
Ref Expression
pwfilem.1 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
Assertion
Ref Expression
pwfilem (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Distinct variable groups:   𝑏,𝑐   𝑥,𝑐
Allowed substitution hints:   𝐹(𝑥,𝑏,𝑐)

Proof of Theorem pwfilem
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 pwundif 5449 . 2 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2 vex 3496 . . . . . . . . 9 𝑐 ∈ V
3 snex 5322 . . . . . . . . 9 {𝑥} ∈ V
42, 3unex 7461 . . . . . . . 8 (𝑐 ∪ {𝑥}) ∈ V
5 pwfilem.1 . . . . . . . 8 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
64, 5fnmpti 6484 . . . . . . 7 𝐹 Fn 𝒫 𝑏
7 dffn4 6589 . . . . . . 7 (𝐹 Fn 𝒫 𝑏𝐹:𝒫 𝑏onto→ran 𝐹)
86, 7mpbi 232 . . . . . 6 𝐹:𝒫 𝑏onto→ran 𝐹
9 fodomfi 8789 . . . . . 6 ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
108, 9mpan2 689 . . . . 5 (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11 domfi 8731 . . . . 5 ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
1210, 11mpdan 685 . . . 4 (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13 eldifi 4101 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
143elpwun 7483 . . . . . . . . 9 (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
1513, 14sylib 220 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16 undif1 4422 . . . . . . . . 9 ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17 elpwunsn 4613 . . . . . . . . . . 11 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥𝑑)
1817snssd 4734 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19 ssequn2 4157 . . . . . . . . . 10 ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
2018, 19sylib 220 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
2116, 20syl5req 2867 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22 uneq1 4130 . . . . . . . . 9 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
2322rspceeqv 3636 . . . . . . . 8 (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2415, 21, 23syl2anc 586 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
255, 4elrnmpti 5825 . . . . . . 7 (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2624, 25sylibr 236 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
2726ssriv 3969 . . . . 5 (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28 ssdomg 8547 . . . . 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 8731 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
3112, 29, 30syl2anc 586 . . 3 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
32 unfi 8777 . . 3 (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
3331, 32mpancom 686 . 2 (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
341, 33eqeltrid 2915 1 (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wrex 3137  cdif 3931  cun 3932  wss 3934  𝒫 cpw 4537  {csn 4559   class class class wbr 5057  cmpt 5137  ran crn 5549   Fn wfn 6343  ontowfo 6346  cdom 8499  Fincfn 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-fin 8505
This theorem is referenced by:  pwfi  8811
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