Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mptsuppdifd | Structured version Visualization version GIF version |
Description: The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.) |
Ref | Expression |
---|---|
mptsuppdifd.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
mptsuppdifd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
mptsuppdifd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
Ref | Expression |
---|---|
mptsuppdifd | ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptsuppdifd.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptsuppdifd.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2 | mptexd 7040 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
4 | 1, 3 | eqeltrid 2842 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | mptsuppdifd.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
6 | suppimacnv 7916 | . . 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 6101 | . 2 ⊢ (◡𝐹 “ (V ∖ {𝑍})) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} |
9 | 7, 8 | eqtrdi 2794 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {crab 3065 Vcvv 3408 ∖ cdif 3863 {csn 4541 ↦ cmpt 5135 ◡ccnv 5550 “ cima 5554 (class class class)co 7213 supp csupp 7903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-supp 7904 |
This theorem is referenced by: mptsuppd 7929 extmptsuppeq 7930 suppssov1 7940 suppss2 7942 suppssfv 7944 |
Copyright terms: Public domain | W3C validator |