Theorem faovcl 47174

Description: Closure law for an operation, analogous to fovcl 7497. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
faovcl.1	⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶

Assertion

Ref	Expression
faovcl	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Proof of Theorem faovcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		faovcl.1	. . 3 ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶
2		ffnaov 47173	. . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
3	2	simprbi 496	. . 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 2730	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐹 = 𝐹)
6		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7		eqidd 2730	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑦 = 𝑦)
8	5, 6, 7	aoveq123d 47152	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
9	8	eleq1d 2813	. . 3 ⊢ (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10		eqidd 2730	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐹 = 𝐹)
11		eqidd 2730	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐴)
12		id 22	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
13	10, 11, 12	aoveq123d 47152	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
14	13	eleq1d 2813	. . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
15	9, 14	rspc2v 3596	. 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)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   × cxp 5629   Fn wfn 6494  ⟶wf 6495   ((caov 47092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-aiota 47059  df-dfat 47093  df-afv 47094  df-aov 47095

This theorem is referenced by: (None)