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Theorem ofres 6090
Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
Hypotheses
Ref Expression
ofres.1 (𝜑𝐹 Fn 𝐴)
ofres.2 (𝜑𝐺 Fn 𝐵)
ofres.3 (𝜑𝐴𝑉)
ofres.4 (𝜑𝐵𝑊)
ofres.5 (𝐴𝐵) = 𝐶
Assertion
Ref Expression
ofres (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))

Proof of Theorem ofres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofres.1 . . 3 (𝜑𝐹 Fn 𝐴)
2 ofres.2 . . 3 (𝜑𝐺 Fn 𝐵)
3 ofres.3 . . 3 (𝜑𝐴𝑉)
4 ofres.4 . . 3 (𝜑𝐵𝑊)
5 ofres.5 . . 3 (𝐴𝐵) = 𝐶
6 eqidd 2178 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2178 . . 3 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 6083 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
9 inss1 3355 . . . . 5 (𝐴𝐵) ⊆ 𝐴
105, 9eqsstrri 3188 . . . 4 𝐶𝐴
11 fnssres 5324 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 413 . . 3 (𝜑 → (𝐹𝐶) Fn 𝐶)
13 inss2 3356 . . . . 5 (𝐴𝐵) ⊆ 𝐵
145, 13eqsstrri 3188 . . . 4 𝐶𝐵
15 fnssres 5324 . . . 4 ((𝐺 Fn 𝐵𝐶𝐵) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 413 . . 3 (𝜑 → (𝐺𝐶) Fn 𝐶)
17 ssexg 4139 . . . 4 ((𝐶𝐴𝐴𝑉) → 𝐶 ∈ V)
1810, 3, 17sylancr 414 . . 3 (𝜑𝐶 ∈ V)
19 inidm 3344 . . 3 (𝐶𝐶) = 𝐶
20 fvres 5534 . . . 4 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
2120adantl 277 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
22 fvres 5534 . . . 4 (𝑥𝐶 → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2322adantl 277 . . 3 ((𝜑𝑥𝐶) → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2412, 16, 18, 18, 19, 21, 23offval 6083 . 2 (𝜑 → ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
258, 24eqtr4d 2213 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cin 3128  wss 3129  cmpt 4061  cres 4624   Fn wfn 5206  cfv 5211  (class class class)co 5868  𝑓 cof 6074
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-of 6076
This theorem is referenced by: (None)
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