Theorem faovcl 47794

Description: Closure law for an operation, analogous to fovcl 7524. (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 47793	. . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
3	2	simprbi 501	. . 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 2763	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐹 = 𝐹)
6		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7		eqidd 2763	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑦 = 𝑦)
8	5, 6, 7	aoveq123d 47772	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
9	8	eleq1d 2847	. . 3 ⊢ (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10		eqidd 2763	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐹 = 𝐹)
11		eqidd 2763	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐴)
12		id 22	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
13	10, 11, 12	aoveq123d 47772	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
14	13	eleq1d 2847	. . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
15	9, 14	rspc2v 3592	. 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 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   × cxp 5645   Fn wfn 6516  ⟶wf 6517   ((caov 47712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-aiota 47679  df-dfat 47713  df-afv 47714  df-aov 47715

This theorem is referenced by: (None)