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Theorem faovcl 41786
Description: Closure law for an operation, analogous to fovcl 6930. (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 41785 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
32simprbi 483 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)
41, 3ax-mp 5 . 2 𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶
5 eqidd 2761 . . . . 5 (𝑥 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
7 eqidd 2761 . . . . 5 (𝑥 = 𝐴𝑦 = 𝑦)
85, 6, 7aoveq123d 41764 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
98eleq1d 2824 . . 3 (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10 eqidd 2761 . . . . 5 (𝑦 = 𝐵𝐹 = 𝐹)
11 eqidd 2761 . . . . 5 (𝑦 = 𝐵𝐴 = 𝐴)
12 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
1310, 11, 12aoveq123d 41764 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
1413eleq1d 2824 . . 3 (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
159, 14rspc2v 3461 . 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 1632  wcel 2139  wral 3050   × cxp 5264   Fn wfn 6044  wf 6045   ((caov 41701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-dfat 41702  df-afv 41703  df-aov 41704
This theorem is referenced by: (None)
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