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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovmpt2x2 | Structured version Visualization version GIF version |
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6790 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.) |
Ref | Expression |
---|---|
ovmpt2x2.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
ovmpt2x2.2 | ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐿) |
ovmpt2x2.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ovmpt2x2 | ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpt2x2.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
3 | ovmpt2x2.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) | |
4 | 3 | adantl 482 | . 2 ⊢ (((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
5 | ovmpt2x2.2 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐿) | |
6 | 5 | adantl 482 | . 2 ⊢ (((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) |
7 | simp1 1061 | . 2 ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → 𝐴 ∈ 𝐿) | |
8 | simp2 1062 | . 2 ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → 𝐵 ∈ 𝐷) | |
9 | simp3 1063 | . 2 ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → 𝑆 ∈ 𝐻) | |
10 | 2, 4, 6, 7, 8, 9 | ovmpt2rdx 42118 | 1 ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = 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: lincval 42198 |
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