Theorem mptsuppdifd 8241

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 7261	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4	1, 3	eqeltrid 2868	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptsuppdifd.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6		suppimacnv 8229	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8	1	mptpreima 6258	. 2 ⊢ (◡𝐹 “ (V ∖ {𝑍})) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}
9	7, 8	eqtrdi 2814	1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ∖ cdif 4048  {csn 4588   ↦ cmpt 5192  ^◡ccnv 5657   “ cima 5661  (class class class)co 7451   supp csupp 8214

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-supp 8215

This theorem is used by:  mptsuppd  8242  extmptsuppeq  8243  suppssov1  8252  suppssov2  8253  suppss2  8255  suppssfv  8257