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Mirrors > Home > MPE Home > Th. List > ovmpodx | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
ovmpodx.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
ovmpodx.2 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
ovmpodx.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿) |
ovmpodx.4 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpodx.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐿) |
ovmpodx.6 | ⊢ (𝜑 → 𝑆 ∈ 𝑋) |
Ref | Expression |
---|---|
ovmpodx | ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpodx.1 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpodx.2 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
3 | ovmpodx.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿) | |
4 | ovmpodx.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
5 | ovmpodx.5 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐿) | |
6 | ovmpodx.6 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋) | |
7 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1917 | . 2 ⊢ Ⅎ𝑦𝜑 | |
9 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐴 | |
10 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐵 | |
11 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝑆 | |
12 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝑆 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpodxf 7538 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 (class class class)co 7390 ∈ cmpo 7392 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6481 df-fun 6531 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 |
This theorem is referenced by: ovmpod 7540 ovmpox 7541 dpjfval 19881 fgval 23298 om1val 24470 pi1val 24477 dvfval 25338 dvnfval 25363 taylfval 25795 line 47052 rrxline 47054 |
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