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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovmpordx | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7509. (Contributed by AV, 30-Mar-2019.) |
Ref | Expression |
---|---|
ovmpordx.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
ovmpordx.2 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
ovmpordx.3 | ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) |
ovmpordx.4 | ⊢ (𝜑 → 𝐴 ∈ 𝐿) |
ovmpordx.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
ovmpordx.6 | ⊢ (𝜑 → 𝑆 ∈ 𝑋) |
Ref | Expression |
---|---|
ovmpordx | ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpordx.1 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpordx.2 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
3 | ovmpordx.3 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) | |
4 | ovmpordx.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐿) | |
5 | ovmpordx.5 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
6 | ovmpordx.6 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋) | |
7 | nfv 1918 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1918 | . 2 ⊢ Ⅎ𝑦𝜑 | |
9 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐴 | |
10 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝐵 | |
11 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝑆 | |
12 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝑆 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpordxf 46504 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7361 ∈ cmpo 7363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 |
This theorem is referenced by: ovmpox2 46506 |
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