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Theorem pwfilem 9354
Description: Lemma for pwfi 9355. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5371. (Revised by BTernaryTau, 7-Sep-2024.)
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 4629 . 2 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2 pwfilem.1 . . . . . 6 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
32funmpt2 6607 . . . . 5 Fun 𝐹
4 vex 3482 . . . . . . . . . 10 𝑐 ∈ V
5 vsnex 5440 . . . . . . . . . 10 {𝑥} ∈ V
64, 5unex 7763 . . . . . . . . 9 (𝑐 ∪ {𝑥}) ∈ V
76, 2dmmpti 6713 . . . . . . . 8 dom 𝐹 = 𝒫 𝑏
87imaeq2i 6078 . . . . . . 7 (𝐹 “ dom 𝐹) = (𝐹 “ 𝒫 𝑏)
9 imadmrn 6090 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
108, 9eqtr3i 2765 . . . . . 6 (𝐹 “ 𝒫 𝑏) = ran 𝐹
11 imafi 9351 . . . . . 6 ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → (𝐹 “ 𝒫 𝑏) ∈ Fin)
1210, 11eqeltrrid 2844 . . . . 5 ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → ran 𝐹 ∈ Fin)
133, 12mpan 690 . . . 4 (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
14 eldifi 4141 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
155elpwun 7788 . . . . . . . 8 (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
1614, 15sylib 218 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
17 undif1 4482 . . . . . . . 8 ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
18 elpwunsn 4689 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥𝑑)
1918snssd 4814 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
20 ssequn2 4199 . . . . . . . . 9 ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
2119, 20sylib 218 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
2217, 21eqtr2id 2788 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23 uneq1 4171 . . . . . . . 8 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
2423rspceeqv 3645 . . . . . . 7 (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2516, 22, 24syl2anc 584 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
262, 25, 14elrnmptd 5977 . . . . 5 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
2726ssriv 3999 . . . 4 (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28 ssfi 9212 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
2913, 27, 28sylancl 586 . . 3 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
30 unfi 9210 . . 3 (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
3129, 30mpancom 688 . 2 (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
321, 31eqeltrid 2843 1 (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960  cun 3961  wss 3963  𝒫 cpw 4605  {csn 4631  cmpt 5231  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  pwfi  9355
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