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Mathbox for Alexander van der Vekens |
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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 7489. (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 45505 | . . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)) | |
3 | 2 | simprbi 498 | . . 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 45484 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) ) |
9 | 8 | eleq1d 2823 | . . 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 45484 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) ) |
14 | 13 | eleq1d 2823 | . . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶)) |
15 | 9, 14 | rspc2v 3593 | . 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 × cxp 5636 Fn wfn 6496 ⟶wf 6497 ((caov 45424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-aiota 45391 df-dfat 45425 df-afv 45426 df-aov 45427 |
This theorem is referenced by: (None) |
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