![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8171 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
5 | eldifsn 4785 | . . . 4 ⊢ (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍)) | |
6 | mptsuppd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) | |
7 | 6 | elexd 3489 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
8 | 7 | biantrurd 532 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≠ 𝑍 ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑍))) |
9 | 5, 8 | bitr4id 290 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵 ≠ 𝑍)) |
10 | 9 | rabbidva 3433 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
11 | 4, 10 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 Vcvv 3468 ∖ cdif 3940 {csn 4623 ↦ cmpt 5224 (class class class)co 7405 supp csupp 8146 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-supp 8147 |
This theorem is referenced by: rmsupp0 47317 domnmsuppn0 47318 rmsuppss 47319 suppmptcfin 47328 lcoc0 47375 linc1 47378 |
Copyright terms: Public domain | W3C validator |