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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 7582. (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 1911 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1911 | . 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 | ovmpordxf 48183 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ∈ cmpo 7432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 |
This theorem is referenced by: ovmpox2 48185 |
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