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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 7583. (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 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 8 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 9 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝐴 | |
| 10 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐵 | |
| 11 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝑆 | |
| 12 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝑆 | |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpordxf 48255 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ∈ cmpo 7433 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: ovmpox2 48257 |
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