Theorem faovcl 47150

Description: Closure law for an operation, analogous to fovcl 7561. (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 47149	. . . 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 2736	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐹 = 𝐹)
6		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7		eqidd 2736	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑦 = 𝑦)
8	5, 6, 7	aoveq123d 47128	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
9	8	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10		eqidd 2736	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐹 = 𝐹)
11		eqidd 2736	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐴)
12		id 22	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
13	10, 11, 12	aoveq123d 47128	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
14	13	eleq1d 2824	. . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
15	9, 14	rspc2v 3633	. 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 1537   ∈ wcel 2106  ∀wral 3059   × cxp 5687   Fn wfn 6558  ⟶wf 6559   ((caov 47068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070  df-aov 47071

This theorem is referenced by: (None)