Theorem faovcl 43406

Description: Closure law for an operation, analogous to fovcl 7281. (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 43405	. . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
3	2	simprbi 499	. . 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 2824	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐹 = 𝐹)
6		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7		eqidd 2824	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑦 = 𝑦)
8	5, 6, 7	aoveq123d 43384	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
9	8	eleq1d 2899	. . 3 ⊢ (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10		eqidd 2824	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐹 = 𝐹)
11		eqidd 2824	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐴)
12		id 22	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
13	10, 11, 12	aoveq123d 43384	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
14	13	eleq1d 2899	. . 3 ⊢ (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
15	9, 14	rspc2v 3635	. 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 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   × cxp 5555   Fn wfn 6352  ⟶wf 6353   ((caov 43324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-aiota 43292  df-dfat 43325  df-afv 43326  df-aov 43327

This theorem is referenced by: (None)