Theorem mptsuppdifd 7835

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 6964	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4	1, 3	eqeltrid 2894	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptsuppdifd.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6		suppimacnv 7824	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 587	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8	1	mptpreima 6059	. 2 ⊢ (◡𝐹 “ (V ∖ {𝑍})) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}
9	7, 8	eqtrdi 2849	1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3878  {csn 4525   ↦ cmpt 5110  ^◡ccnv 5518   “ cima 5522  (class class class)co 7135   supp csupp 7813

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-supp 7814

This theorem is referenced by:  mptsuppd  7836  extmptsuppeq  7837  suppssov1  7845  suppss2  7847  suppssfv  7849