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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 7949 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
5 | eldifsn 4715 | . . . 4 ⊢ (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍)) | |
6 | mptsuppd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) | |
7 | 6 | elexd 3441 | . . . . 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 3401 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
11 | 4, 10 | eqtrd 2778 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 {crab 3066 Vcvv 3421 ∖ cdif 3878 {csn 4556 ↦ cmpt 5150 (class class class)co 7232 supp csupp 7924 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-supp 7925 |
This theorem is referenced by: rmsupp0 45408 domnmsuppn0 45409 rmsuppss 45410 suppmptcfin 45419 lcoc0 45467 linc1 45470 |
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