Theorem ofmres 5921

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 5922, allowing it to be used as a function or structure argument. By ofmresval 5881, 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 3047	. . 3 ⊢ 𝐴 ⊆ V
2		ssv 3047	. . 3 ⊢ 𝐵 ⊆ V
3		resmpt2 5757	. . 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 418	. 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 5870	. . 3 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6	5	reseq1i 4722	. 2 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵))
7		eqid 2089	. . 3 ⊢ 𝐴 = 𝐴
8		eqid 2089	. . 3 ⊢ 𝐵 = 𝐵
9		vex 2623	. . . 4 ⊢ 𝑓 ∈ V
10		vex 2623	. . . 4 ⊢ 𝑔 ∈ V
11	9	dmex 4712	. . . . . 6 ⊢ dom 𝑓 ∈ V
12	11	inex1 3979	. . . . 5 ⊢ (dom 𝑓 ∩ dom 𝑔) ∈ V
13	12	mptex 5537	. . . 4 ⊢ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V
14	5	ovmpt4g 5781	. . . 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 1274	. . 3 ⊢ (𝑓 ∘𝑓 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))
16	7, 8, 15	mpt2eq123i 5726	. 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 2119	1 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔))

Syntax hints:   = wceq 1290   ∈ wcel 1439  Vcvv 2620   ∩ cin 2999   ⊆ wss 3000   ↦ cmpt 3905   × cxp 4450  dom cdm 4452   ↾ cres 4454  ‘cfv 5028  (class class class)co 5666   ↦ cmpt2 5668   ∘_𝑓 cof 5868

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-of 5870

This theorem is referenced by: (None)