Theorem foprcl 4021

Description: Closure law for an operation.

Hypothesis

Ref	Expression
foprcl.1	⊢ F:(R × S)–→C

Assertion

Ref	Expression
foprcl	⊢ ((A ∈ R ⋀ B ∈ S) → (AFB) ∈ C)

Proof of Theorem foprcl

Step	Hyp	Ref	Expression
1		foprcl.1	. . 3 ⊢ F:(R × S)–→C
2		ffnoprval 4020	. . . 4 ⊢ (F:(R × S)–→C ↔ (F Fn (R × S) ⋀ ∀x ∈ R ∀y ∈ S (xFy) ∈ C))
3	2	pm3.27bi 326	. . 3 ⊢ (F:(R × S)–→C → ∀x ∈ R ∀y ∈ S (xFy) ∈ C)
4	1, 3	ax-mp 7	. 2 ⊢ ∀x ∈ R ∀y ∈ S (xFy) ∈ C
5		opreq1 3974	. . . 4 ⊢ (x = A → (xFy) = (AFy))
6	5	eleq1d 1543	. . 3 ⊢ (x = A → ((xFy) ∈ C ↔ (AFy) ∈ C))
7		opreq2 3975	. . . 4 ⊢ (y = B → (AFy) = (AFB))
8	7	eleq1d 1543	. . 3 ⊢ (y = B → ((AFy) ∈ C ↔ (AFB) ∈ C))
9	6, 8	rcla42v 1883	. 2 ⊢ ((A ∈ R ⋀ B ∈ S) → (∀x ∈ R ∀y ∈ S (xFy) ∈ C → (AFB) ∈ C))
10	4, 9	mpi 44	1 ⊢ ((A ∈ R ⋀ B ∈ S) → (AFB) ∈ C)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648   × cxp 3174   Fn wfn 3183  –→wf 3184  (class class class)co 3969

This theorem is referenced by:  axaddcl 5283  axmulcl 5285  issubgi 8118  ablmul 8127  hvaddclt 8877  hvmulclt 8878  hiclt 8942  iooirrsa 10478

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971

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