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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovmpt2rdx | 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 ovmpt2dxf 6786. (Contributed by AV, 30-Mar-2019.) |
Ref | Expression |
---|---|
ovmpt2rdx.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
ovmpt2rdx.2 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
ovmpt2rdx.3 | ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) |
ovmpt2rdx.4 | ⊢ (𝜑 → 𝐴 ∈ 𝐿) |
ovmpt2rdx.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
ovmpt2rdx.6 | ⊢ (𝜑 → 𝑆 ∈ 𝑋) |
Ref | Expression |
---|---|
ovmpt2rdx | ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpt2rdx.1 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpt2rdx.2 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
3 | ovmpt2rdx.3 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) | |
4 | ovmpt2rdx.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐿) | |
5 | ovmpt2rdx.5 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
6 | ovmpt2rdx.6 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋) | |
7 | nfv 1843 | . 2 ⊢ Ⅎ𝑥𝜑 | |
8 | nfv 1843 | . 2 ⊢ Ⅎ𝑦𝜑 | |
9 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐴 | |
10 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐵 | |
11 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝑆 | |
12 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝑆 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ovmpt2rdxf 42117 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: ovmpt2x2 42119 |
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