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Theorem offval2 5949
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1 (𝜑𝐴𝑉)
offval2.2 ((𝜑𝑥𝐴) → 𝐵𝑊)
offval2.3 ((𝜑𝑥𝐴) → 𝐶𝑋)
offval2.4 (𝜑𝐹 = (𝑥𝐴𝐵))
offval2.5 (𝜑𝐺 = (𝑥𝐴𝐶))
Assertion
Ref Expression
offval2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem offval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑊)
21ralrimiva 2477 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
3 eqid 2113 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 5205 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 offval2.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 5169 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 166 . . 3 (𝜑𝐹 Fn 𝐴)
9 offval2.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝑋)
109ralrimiva 2477 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
11 eqid 2113 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1211fnmpt 5205 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1310, 12syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
14 offval2.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1514fneq1d 5169 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1613, 15mpbird 166 . . 3 (𝜑𝐺 Fn 𝐴)
17 offval2.1 . . 3 (𝜑𝐴𝑉)
18 inidm 3249 . . 3 (𝐴𝐴) = 𝐴
196adantr 272 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2019fveq1d 5375 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2114adantr 272 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2221fveq1d 5375 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
238, 16, 17, 17, 18, 20, 22offval 5941 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
24 nffvmpt1 5384 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
25 nfcv 2253 . . . . 5 𝑥𝑅
26 nffvmpt1 5384 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
2724, 25, 26nfov 5753 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
28 nfcv 2253 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
29 fveq2 5373 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
30 fveq2 5373 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3129, 30oveq12d 5744 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3227, 28, 31cbvmpt 3981 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
33 simpr 109 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
343fvmpt2 5456 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 1, 34syl2anc 406 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3611fvmpt2 5456 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3733, 9, 36syl2anc 406 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3835, 37oveq12d 5744 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
3938mpteq2dva 3976 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4032, 39syl5eq 2157 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4123, 40eqtrd 2145 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461  wral 2388  cmpt 3947   Fn wfn 5074  cfv 5079  (class class class)co 5726  𝑓 cof 5932
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-of 5934
This theorem is referenced by:  ofc12  5954  caofinvl  5956  caofcom  5957
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