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Mirrors > Home > MPE Home > Th. List > Mathboxes > faovcl | Structured version Visualization version GIF version |
Description: Closure law for an operation, analogous to fovcl 7338. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
faovcl.1 | ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 |
Ref | Expression |
---|---|
faovcl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | faovcl.1 | . . 3 ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 | |
2 | ffnaov 44363 | . . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)) | |
3 | 2 | simprbi 500 | . . 3 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶 |
5 | eqidd 2738 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐹 = 𝐹) | |
6 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
7 | eqidd 2738 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑦 = 𝑦) | |
8 | 5, 6, 7 | aoveq123d 44342 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) ) |
9 | 8 | eleq1d 2822 | . . 3 ⊢ (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶)) |
10 | eqidd 2738 | . . . . 5 ⊢ (𝑦 = 𝐵 → 𝐹 = 𝐹) | |
11 | eqidd 2738 | . . . . 5 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐴) | |
12 | id 22 | . . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
13 | 10, 11, 12 | aoveq123d 44342 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) ) |
14 | 13 | eleq1d 2822 | . . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶)) |
15 | 9, 14 | rspc2v 3547 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶 → ((𝐴𝐹𝐵)) ∈ 𝐶)) |
16 | 4, 15 | mpi 20 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 × cxp 5549 Fn wfn 6375 ⟶wf 6376 ((caov 44282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-aiota 44249 df-dfat 44283 df-afv 44284 df-aov 44285 |
This theorem is referenced by: (None) |
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