Theorem pwfilem 8502

Description: Lemma for pwfi 8503. (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 5217	. 2 ⊢ 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2		vex 3388	. . . . . . . . 9 ⊢ 𝑐 ∈ V
3		snex 5099	. . . . . . . . 9 ⊢ {𝑥} ∈ V
4	2, 3	unex 7190	. . . . . . . 8 ⊢ (𝑐 ∪ {𝑥}) ∈ V
5		pwfilem.1	. . . . . . . 8 ⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
6	4, 5	fnmpti 6233	. . . . . . 7 ⊢ 𝐹 Fn 𝒫 𝑏
7		dffn4 6337	. . . . . . 7 ⊢ (𝐹 Fn 𝒫 𝑏 ↔ 𝐹:𝒫 𝑏–onto→ran 𝐹)
8	6, 7	mpbi 222	. . . . . 6 ⊢ 𝐹:𝒫 𝑏–onto→ran 𝐹
9		fodomfi 8481	. . . . . 6 ⊢ ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏–onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
10	8, 9	mpan2 683	. . . . 5 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11		domfi 8423	. . . . 5 ⊢ ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
12	10, 11	mpdan 679	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13		eldifi 3930	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
14	3	elpwun 7211	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
15	13, 14	sylib 210	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16		undif1 4237	. . . . . . . . 9 ⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17		elpwunsn 4415	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑)
18	17	snssd 4528	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19		ssequn2 3984	. . . . . . . . . 10 ⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
20	18, 19	sylib 210	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
21	16, 20	syl5req 2846	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22		uneq1 3958	. . . . . . . . 9 ⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23	22	rspceeqv 3515	. . . . . . . 8 ⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
24	15, 21, 23	syl2anc 580	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
25	5, 4	elrnmpti 5580	. . . . . . 7 ⊢ (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
26	24, 25	sylibr 226	. . . . . 6 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
27	26	ssriv 3802	. . . . 5 ⊢ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28		ssdomg 8241	. . . . 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 8423	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
31	12, 29, 30	syl2anc 580	. . 3 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
32		unfi 8469	. . 3 ⊢ (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
33	31, 32	mpancom 680	. 2 ⊢ (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
34	1, 33	syl5eqel 2882	1 ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  ∃wrex 3090   ∖ cdif 3766   ∪ cun 3767   ⊆ wss 3769  𝒫 cpw 4349  {csn 4368   class class class wbr 4843   ↦ cmpt 4922  ran crn 5313   Fn wfn 6096  –onto→wfo 6099   ≼ cdom 8193  Fincfn 8195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-fin 8199

This theorem is referenced by:  pwfi  8503