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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 7286. (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 1915 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜑 | |
9 | nfcv 2977 | . 2 ⊢ Ⅎ𝑦𝐴 | |
10 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝐵 | |
11 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝑆 | |
12 | nfcv 2977 | . 2 ⊢ Ⅎ𝑦𝑆 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpordxf 44472 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7142 ∈ cmpo 7144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 |
This theorem is referenced by: ovmpox2 44474 |
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