Theorem mptsuppdifd 8002

Description: The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)

Hypotheses

Ref	Expression
mptsuppdifd.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
mptsuppdifd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mptsuppdifd.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)

Assertion

Ref	Expression
mptsuppdifd	⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})

Distinct variable groups:   𝑥,𝐴   𝑥,𝑍

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptsuppdifd

Step	Hyp	Ref	Expression
1		mptsuppdifd.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptsuppdifd.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	mptexd 7100	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4	1, 3	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptsuppdifd.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6		suppimacnv 7990	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8	1	mptpreima 6141	. 2 ⊢ (◡𝐹 “ (V ∖ {𝑍})) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}
9	7, 8	eqtrdi 2794	1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432   ∖ cdif 3884  {csn 4561   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592  (class class class)co 7275   supp csupp 7977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978

This theorem is referenced by:  mptsuppd  8003  extmptsuppeq  8004  suppssov1  8014  suppss2  8016  suppssfv  8018