Theorem pwfilem 9221

Description: Lemma for pwfi 9222. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5302. (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 4566	. 2 ⊢ 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2		pwfilem.1	. . . . . 6 ⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
3	2	funmpt2 6531	. . . . 5 ⊢ Fun 𝐹
4		vex 3434	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
5		vsnex 5372	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
6	4, 5	unex 7691	. . . . . . . . 9 ⊢ (𝑐 ∪ {𝑥}) ∈ V
7	6, 2	dmmpti 6636	. . . . . . . 8 ⊢ dom 𝐹 = 𝒫 𝑏
8	7	imaeq2i 6017	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = (𝐹 “ 𝒫 𝑏)
9		imadmrn 6029	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
10	8, 9	eqtr3i 2762	. . . . . 6 ⊢ (𝐹 “ 𝒫 𝑏) = ran 𝐹
11		imafi 9218	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → (𝐹 “ 𝒫 𝑏) ∈ Fin)
12	10, 11	eqeltrrid 2842	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝒫 𝑏 ∈ Fin) → ran 𝐹 ∈ Fin)
13	3, 12	mpan 691	. . . 4 ⊢ (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
14		eldifi 4072	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
15	5	elpwun 7716	. . . . . . . 8 ⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16	14, 15	sylib 218	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
17		undif1 4417	. . . . . . . 8 ⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
18		elpwunsn 4629	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑)
19	18	snssd 4753	. . . . . . . . 9 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
20		ssequn2 4130	. . . . . . . . 9 ⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
21	19, 20	sylib 218	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
22	17, 21	eqtr2id 2785	. . . . . . 7 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
23		uneq1 4102	. . . . . . . 8 ⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
24	23	rspceeqv 3588	. . . . . . 7 ⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
25	16, 22, 24	syl2anc 585	. . . . . 6 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
26	2, 25, 14	elrnmptd 5912	. . . . 5 ⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
27	26	ssriv 3926	. . . 4 ⊢ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
28		ssfi 9100	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
29	13, 27, 28	sylancl 587	. . 3 ⊢ (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
30		unfi 9098	. . 3 ⊢ (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
31	29, 30	mpancom 689	. 2 ⊢ (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
32	1, 31	eqeltrid 2841	1 ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568   ↦ cmpt 5167  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486  Fincfn 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7811  df-1o 8398  df-en 8887  df-dom 8888  df-fin 8890

This theorem is referenced by:  pwfi  9222