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Theorem faovcl 39929
 Description: Closure law for an operation, analogous to fovcl 6663. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
faovcl.1 𝐹:(𝑅 × 𝑆)⟶𝐶
Assertion
Ref Expression
faovcl ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Proof of Theorem faovcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 faovcl.1 . . 3 𝐹:(𝑅 × 𝑆)⟶𝐶
2 ffnaov 39928 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
32simprbi 479 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)
41, 3ax-mp 5 . 2 𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶
5 eqidd 2611 . . . . 5 (𝑥 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
7 eqidd 2611 . . . . 5 (𝑥 = 𝐴𝑦 = 𝑦)
85, 6, 7aoveq123d 39907 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
98eleq1d 2672 . . 3 (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10 eqidd 2611 . . . . 5 (𝑦 = 𝐵𝐹 = 𝐹)
11 eqidd 2611 . . . . 5 (𝑦 = 𝐵𝐴 = 𝐴)
12 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
1310, 11, 12aoveq123d 39907 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
1413eleq1d 2672 . . 3 (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
159, 14rspc2v 3293 . 2 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶 → ((𝐴𝐹𝐵)) ∈ 𝐶))
164, 15mpi 20 1 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036   Fn wfn 5799  ⟶wf 5800   ((caov 39844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847 This theorem is referenced by: (None)
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