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Mirrors > Home > MPE Home > Th. List > mptsuppd | Structured version Visualization version GIF version |
Description: The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
mptsuppdifd.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
mptsuppdifd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
mptsuppdifd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
mptsuppd.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
mptsuppd | ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptsuppdifd.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptsuppdifd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | mptsuppdifd.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
4 | 1, 2, 3 | mptsuppdifd 8199 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
5 | eldifsn 4795 | . . . 4 ⊢ (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍)) | |
6 | mptsuppd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) | |
7 | 6 | elexd 3494 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
8 | 7 | biantrurd 531 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≠ 𝑍 ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍))) |
9 | 5, 8 | bitr4id 289 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵 ≠ 𝑍)) |
10 | 9 | rabbidva 3437 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
11 | 4, 10 | eqtrd 2768 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 {crab 3430 Vcvv 3473 ∖ cdif 3946 {csn 4632 ↦ cmpt 5235 (class class class)co 7426 supp csupp 8173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-supp 8174 |
This theorem is referenced by: rmsupp0 47528 domnmsuppn0 47529 rmsuppss 47530 suppmptcfin 47539 lcoc0 47586 linc1 47589 |
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