Theorem ofmres 7928

Description: Equivalent expressions for a restriction of the function operation map. Unlike ∘_f 𝑅 which is a proper class, ( ∘_f 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 7929, allowing it to be used as a function or structure argument. By ofmresval 7638, the restricted operation map values are the same as the original values, allowing theorems for ∘_f 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)

Assertion

Ref	Expression
ofmres	⊢ ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘f 𝑅𝑔))

Distinct variable groups:   𝑓,𝑔,𝐴   𝐵,𝑓,𝑔   𝑅,𝑓,𝑔

Proof of Theorem ofmres

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3958	. . 3 ⊢ 𝐴 ⊆ V
2		ssv 3958	. . 3 ⊢ 𝐵 ⊆ V
3		resmpo 7478	. . 3 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))))
4	1, 2, 3	mp2an 692	. 2 ⊢ ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
5		df-of 7622	. . 3 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6	5	reseq1i 5934	. 2 ⊢ ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵))
7		eqid 2736	. . 3 ⊢ 𝐴 = 𝐴
8		eqid 2736	. . 3 ⊢ 𝐵 = 𝐵
9		vex 3444	. . . 4 ⊢ 𝑓 ∈ V
10		vex 3444	. . . 4 ⊢ 𝑔 ∈ V
11	9	dmex 7851	. . . . . 6 ⊢ dom 𝑓 ∈ V
12	11	inex1 5262	. . . . 5 ⊢ (dom 𝑓 ∩ dom 𝑔) ∈ V
13	12	mptex 7169	. . . 4 ⊢ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V
14	5	ovmpt4g 7505	. . . 4 ⊢ ((𝑓 ∈ V ∧ 𝑔 ∈ V ∧ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V) → (𝑓 ∘f 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
15	9, 10, 13, 14	mp3an 1463	. . 3 ⊢ (𝑓 ∘f 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))
16	7, 8, 15	mpoeq123i 7434	. 2 ⊢ (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘f 𝑅𝑔)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
17	4, 6, 16	3eqtr4i 2769	1 ⊢ ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘f 𝑅𝑔))

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   ↦ cmpt 5179   × cxp 5622  dom cdm 5624   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ∘_f cof 7620

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622

This theorem is referenced by:  mplsubrglem  21959  psrplusgpropd  22176  ofoafg  43596  naddcnff  43604