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Theorem faovcl 41778
Description: Closure law for an operation, analogous to fovcl 6922. (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 41777 . . . 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 2753 . . . . 5 (𝑥 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
7 eqidd 2753 . . . . 5 (𝑥 = 𝐴𝑦 = 𝑦)
85, 6, 7aoveq123d 41756 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
98eleq1d 2816 . . 3 (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10 eqidd 2753 . . . . 5 (𝑦 = 𝐵𝐹 = 𝐹)
11 eqidd 2753 . . . . 5 (𝑦 = 𝐵𝐴 = 𝐴)
12 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
1310, 11, 12aoveq123d 41756 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
1413eleq1d 2816 . . 3 (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
159, 14rspc2v 3453 . 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 1624  wcel 2131  wral 3042   × cxp 5256   Fn wfn 6036  wf 6037   ((caov 41693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-dfat 41694  df-afv 41695  df-aov 41696
This theorem is referenced by: (None)
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