Theorem isf34lem1 9232

Description: Lemma for isfin3-4 9242. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
compss.a	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))

Assertion

Ref	Expression
isf34lem1	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem isf34lem1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw2g 4857	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
2	1	biimpar 501	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
3		difexg 4841	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑋) ∈ V)
4	3	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∖ 𝑋) ∈ V)
5		difeq2 3755	. . 3 ⊢ (𝑎 = 𝑋 → (𝐴 ∖ 𝑎) = (𝐴 ∖ 𝑋))
6		compss.a	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
7		difeq2 3755	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
8	7	cbvmptv 4783	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
9	6, 8	eqtri 2673	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
10	5, 9	fvmptg 6319	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑋) ∈ V) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))
11	2, 4, 10	syl2anc 694	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  𝒫 cpw 4191   ↦ cmpt 4762  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934

This theorem is referenced by:  compssiso  9234  isf34lem4  9237  isf34lem7  9239  isf34lem6  9240