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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovmpogad | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7581. (Contributed by SN, 14-Mar-2025.) |
Ref | Expression |
---|---|
ovmpogad.f | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
ovmpogad.s | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
ovmpogad.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpogad.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
ovmpogad.v | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
Ref | Expression |
---|---|
ovmpogad | ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpogad.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
3 | ovmpogad.s | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) | |
4 | 3 | adantl 480 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
5 | ovmpogad.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
6 | ovmpogad.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
7 | ovmpogad.v | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
8 | 2, 4, 5, 6, 7 | ovmpod 7579 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7426 ∈ cmpo 7428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 |
This theorem is referenced by: (None) |
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