Theorem pwfilem 8960

Description: Lemma for pwfi 8961. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5288. (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.

Step	Hyp	Ref	Expression
1		pwundif 4559	. 2 ⊢ 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2		pwfilem.1	. . . . . 6 ⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
3	2	funmpt2 6473	. . . . 5 ⊢ Fun 𝐹
4		vex 3436	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
5		snex 5354	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
6	4, 5	unex 7596	. . . . . . . . 9 ⊢ (𝑐 ∪ {𝑥}) ∈ V
7	6, 2	dmmpti 6577	. . . . . . . 8 ⊢ dom 𝐹 = 𝒫 𝑏
8	7	imaeq2i 5967	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = (𝐹 “ 𝒫 𝑏)
9		imadmrn 5979	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
10	8, 9	eqtr3i 2768	. . . . . 6 ⊢ (𝐹 “ 𝒫 𝑏) = ran 𝐹
11		imafi 8958	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → (𝐹 “ 𝒫 𝑏) ∈ Fin)
12	10, 11	eqeltrrid 2844	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → ran 𝐹 ∈ Fin)
13	3, 12	mpan 687	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
14		eldifi 4061	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
15	5	elpwun 7619	. . . . . . . 8 ⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16	14, 15	sylib 217	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
17		undif1 4409	. . . . . . . 8 ⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
18		elpwunsn 4619	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑)
19	18	snssd 4742	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
20		ssequn2 4117	. . . . . . . . 9 ⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
21	19, 20	sylib 217	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
22	17, 21	eqtr2id 2791	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23		uneq1 4090	. . . . . . . 8 ⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
24	23	rspceeqv 3575	. . . . . . 7 ⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
25	16, 22, 24	syl2anc 584	. . . . . 6 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
26	2, 25, 14	elrnmptd 5870	. . . . 5 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
27	26	ssriv 3925	. . . 4 ⊢ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28		ssfi 8956	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
29	13, 27, 28	sylancl 586	. . 3 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
30		unfi 8955	. . 3 ⊢ (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
31	29, 30	mpancom 685	. 2 ⊢ (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
32	1, 31	eqeltrid 2843	1 ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   “ cima 5592  Fun wfun 6427  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-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-en 8734  df-fin 8737

This theorem is referenced by:  pwfi  8961  pwfiOLD  9114