Theorem fovcld 6193

Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)

Hypothesis

Ref	Expression
fovcld.1	⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)

Assertion

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

Proof of Theorem fovcld

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

Step	Hyp	Ref	Expression
1		3simpc 1027	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
2		fovcld.1	. . . 4 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)
3		ffnov 6192	. . . . 5 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶))
4	3	simprbi 275	. . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
5	2, 4	syl 14	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
6	5	3ad2ant1 1049	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
7		oveq1 6092	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
8	7	eleq1d 2307	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶))
9		oveq2 6093	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
10	9	eleq1d 2307	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
11	8, 10	rspc2v 2943	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶 → (𝐴𝐹𝐵) ∈ 𝐶))
12	1, 6, 11	sylc 62	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528   × cxp 4772   Fn wfn 5372  ⟶wf 5373  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088

This theorem is used by:  fovcl  6194  imasrng  14255