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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 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
9 | nfcv 2976 | . 2 ⊢ Ⅎ𝑦𝐴 | |
10 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐵 | |
11 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝑆 | |
12 | nfcv 2976 | . 2 ⊢ Ⅎ𝑦𝑆 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpodxf 7293 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ∈ cmpo 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: ovmpod 7295 ovmpox 7296 dpjfval 19170 fgval 22471 om1val 23627 pi1val 23634 dvfval 24490 dvnfval 24514 taylfval 24943 line 44789 rrxline 44791 |
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