Theorem isf34lem1 10294

Description: Lemma for isfin3-4 10304. (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		compss.a	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
2		difeq2 4074	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
3	2	cbvmptv 5204	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
4	1, 3	eqtri 2760	. 2 ⊢ 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
5		difeq2 4074	. 2 ⊢ (𝑎 = 𝑋 → (𝐴 ∖ 𝑎) = (𝐴 ∖ 𝑋))
6		elpw2g 5280	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
7	6	biimpar 477	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
8		difexg 5276	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑋) ∈ V)
9	8	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∖ 𝑋) ∈ V)
10	4, 5, 7, 9	fvmptd3 6973	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4556   ↦ cmpt 5181  ‘cfv 6500

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 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508

This theorem is referenced by:  compssiso  10296  isf34lem4  10299  isf34lem7  10301  isf34lem6  10302