Theorem fovcld 6024

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 998	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
2		fovcld.1	. . . 4 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)
3		ffnov 6023	. . . . 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 1020	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
7		oveq1 5926	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
8	7	eleq1d 2262	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶))
9		oveq2 5927	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
10	9	eleq1d 2262	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
11	8, 10	rspc2v 2878	. 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)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472   × cxp 4658   Fn wfn 5250  ⟶wf 5251  (class class class)co 5919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922

This theorem is referenced by:  fovcl  6025  imasrng  13455