Theorem isf34lem1 10399

Description: Lemma for isfin3-4 10409. (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 4068	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
3	2	cbvmptv 5209	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
4	1, 3	eqtri 2783	. 2 ⊢ 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑎))
5		difeq2 4068	. 2 ⊢ (𝑎 = 𝑋 → (𝐴 ∖ 𝑎) = (𝐴 ∖ 𝑋))
6		elpw2g 5298	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
7	6	biimpar 483	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
8		difexg 5294	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑋) ∈ V)
9	8	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∖ 𝑋) ∈ V)
10	4, 5, 7, 9	fvmptd3 7013	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4557   ↦ cmpt 5186  ‘cfv 6535

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6491  df-fun 6537  df-fv 6543

This theorem is used by:  compssiso  10401  isf34lem4  10404  isf34lem7  10406  isf34lem6  10407