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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 7835 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
5 | eldifsn 4680 | . . . 4 ⊢ (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍)) | |
6 | mptsuppd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) | |
7 | 6 | elexd 3461 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
8 | 7 | biantrurd 536 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≠ 𝑍 ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍))) |
9 | 5, 8 | bitr4id 293 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵 ≠ 𝑍)) |
10 | 9 | rabbidva 3425 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
11 | 4, 10 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ∖ cdif 3878 {csn 4525 ↦ cmpt 5110 (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: rmsupp0 44770 domnmsuppn0 44771 rmsuppss 44772 suppmptcfin 44781 lcoc0 44831 linc1 44834 |
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