Theorem pwfilem 8820

Description: Lemma for pwfi 8821. (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.

Step	Hyp	Ref	Expression
1		pwundif 4567	. 2 ⊢ 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2		vex 3499	. . . . . . . . 9 ⊢ 𝑐 ∈ V
3		snex 5334	. . . . . . . . 9 ⊢ {𝑥} ∈ V
4	2, 3	unex 7471	. . . . . . . 8 ⊢ (𝑐 ∪ {𝑥}) ∈ V
5		pwfilem.1	. . . . . . . 8 ⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
6	4, 5	fnmpti 6493	. . . . . . 7 ⊢ 𝐹 Fn 𝒫 𝑏
7		dffn4 6598	. . . . . . 7 ⊢ (𝐹 Fn 𝒫 𝑏 ↔ 𝐹:𝒫 𝑏–onto→ran 𝐹)
8	6, 7	mpbi 232	. . . . . 6 ⊢ 𝐹:𝒫 𝑏–onto→ran 𝐹
9		fodomfi 8799	. . . . . 6 ⊢ ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏–onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
10	8, 9	mpan2 689	. . . . 5 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11		domfi 8741	. . . . 5 ⊢ ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
12	10, 11	mpdan 685	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13		eldifi 4105	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
14	3	elpwun 7493	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
15	13, 14	sylib 220	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16		undif1 4426	. . . . . . . . 9 ⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17		elpwunsn 4623	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑)
18	17	snssd 4744	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19		ssequn2 4161	. . . . . . . . . 10 ⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
20	18, 19	sylib 220	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
21	16, 20	syl5req 2871	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22		uneq1 4134	. . . . . . . . 9 ⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23	22	rspceeqv 3640	. . . . . . . 8 ⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
24	15, 21, 23	syl2anc 586	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
25	5, 4	elrnmpti 5834	. . . . . . 7 ⊢ (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
26	24, 25	sylibr 236	. . . . . 6 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
27	26	ssriv 3973	. . . . 5 ⊢ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28		ssdomg 8557	. . . . 5 ⊢ (ran 𝐹 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹 → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹))
29	12, 27, 28	mpisyl 21	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹)
30		domfi 8741	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
31	12, 29, 30	syl2anc 586	. . 3 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
32		unfi 8787	. . 3 ⊢ (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
33	31, 32	mpancom 686	. 2 ⊢ (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
34	1, 33	eqeltrid 2919	1 ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  𝒫 cpw 4541  {csn 4569   class class class wbr 5068   ↦ cmpt 5148  ran crn 5558   Fn wfn 6352  –onto→wfo 6355   ≼ cdom 8509  Fincfn 8511

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-fin 8515

This theorem is referenced by:  pwfi  8821