Theorem mptsuppdifd 8126

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 7168	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4	1, 3	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptsuppdifd.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6		suppimacnv 8114	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 590	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8	1	mptpreima 6189	. 2 ⊢ (◡𝐹 “ (V ∖ {𝑍})) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}
9	7, 8	eqtrdi 2790	1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {crab 3391  Vcvv 3431   ∖ cdif 3880  {csn 4555   ↦ cmpt 5153  ^◡ccnv 5617   “ cima 5621  (class class class)co 7356   supp csupp 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-supp 8101

This theorem is referenced by:  mptsuppd  8127  extmptsuppeq  8128  suppssov1  8137  suppssov2  8138  suppss2  8140  suppssfv  8142