Theorem ofmres 6359

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 6360, allowing it to be used as a function or structure argument. By ofmresval 6304, 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 3270	. . 3 ⊢ 𝐴 ⊆ V
2		ssv 3270	. . 3 ⊢ 𝐵 ⊆ V
3		resmpo 6176	. . 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 430	. 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 6292	. . 3 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6	5	reseq1i 5054	. 2 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵))
7		eqid 2238	. . 3 ⊢ 𝐴 = 𝐴
8		eqid 2238	. . 3 ⊢ 𝐵 = 𝐵
9		vex 2824	. . . 4 ⊢ 𝑓 ∈ V
10		vex 2824	. . . 4 ⊢ 𝑔 ∈ V
11	9	dmex 5044	. . . . . 6 ⊢ dom 𝑓 ∈ V
12	11	inex1 4262	. . . . 5 ⊢ (dom 𝑓 ∩ dom 𝑔) ∈ V
13	12	mptex 5934	. . . 4 ⊢ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V
14	5	ovmpt4g 6201	. . . 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 1378	. . 3 ⊢ (𝑓 ∘𝑓 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))
16	7, 8, 15	mpoeq123i 6141	. 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 2269	1 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔))

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220   ↦ cmpt 4187   × cxp 4767  dom cdm 4769   ↾ cres 4771  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077   ∘_𝑓 cof 6290

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-in1 623  ax-in2 624  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-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292

This theorem is referenced by: (None)