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Theorem faovcl 45506
Description: Closure law for an operation, analogous to fovcl 7489. (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 45505 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
32simprbi 498 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)
41, 3ax-mp 5 . 2 𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶
5 eqidd 2738 . . . . 5 (𝑥 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
7 eqidd 2738 . . . . 5 (𝑥 = 𝐴𝑦 = 𝑦)
85, 6, 7aoveq123d 45484 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
98eleq1d 2823 . . 3 (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10 eqidd 2738 . . . . 5 (𝑦 = 𝐵𝐹 = 𝐹)
11 eqidd 2738 . . . . 5 (𝑦 = 𝐵𝐴 = 𝐴)
12 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
1310, 11, 12aoveq123d 45484 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
1413eleq1d 2823 . . 3 (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
159, 14rspc2v 3593 . 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 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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