Theorem pwfilem 9280

Description: Lemma for pwfi 9281. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5338. (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 4589	. 2 ⊢ 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2		pwfilem.1	. . . . . 6 ⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
3	2	funmpt2 6579	. . . . 5 ⊢ Fun 𝐹
4		vex 3461	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
5		vsnex 5408	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
6	4, 5	unex 7748	. . . . . . . . 9 ⊢ (𝑐 ∪ {𝑥}) ∈ V
7	6, 2	dmmpti 6683	. . . . . . . 8 ⊢ dom 𝐹 = 𝒫 𝑏
8	7	imaeq2i 6062	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = (𝐹 “ 𝒫 𝑏)
9		imadmrn 6074	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
10	8, 9	eqtr3i 2790	. . . . . 6 ⊢ (𝐹 “ 𝒫 𝑏) = ran 𝐹
11		imafi 9278	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → (𝐹 “ 𝒫 𝑏) ∈ Fin)
12	10, 11	eqeltrrid 2870	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → ran 𝐹 ∈ Fin)
13	3, 12	mpan 703	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
14		eldifi 4085	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
15	5	elpwun 7770	. . . . . . . 8 ⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16	14, 15	sylib 221	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
17		undif1 4437	. . . . . . . 8 ⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
18		elpwunsn 4652	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑)
19	18	snssd 4754	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
20		ssequn2 4142	. . . . . . . . 9 ⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
21	19, 20	sylib 221	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
22	17, 21	eqtr2id 2813	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23		uneq1 4115	. . . . . . . 8 ⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
24	23	rspceeqv 3606	. . . . . . 7 ⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
25	16, 22, 24	syl2anc 596	. . . . . 6 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
26	2, 25, 14	elrnmptd 5955	. . . . 5 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
27	26	ssriv 3942	. . . 4 ⊢ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28		ssfi 9160	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
29	13, 27, 28	sylancl 598	. . 3 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
30		unfi 9158	. . 3 ⊢ (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
31	29, 30	mpancom 701	. 2 ⊢ (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
32	1, 31	eqeltrid 2869	1 ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  𝒫 cpw 4564  {csn 4591   ↦ cmpt 5194  dom cdm 5663  ran crn 5664   “ cima 5666  Fun wfun 6534  Fincfn 8945

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7865  df-1o 8455  df-en 8946  df-dom 8947  df-fin 8949

This theorem is used by:  pwfi  9281