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Theorem offval2 5779
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 2440 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
3 eqid 2083 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 5077 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 offval2.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 5041 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 165 . . 3 (𝜑𝐹 Fn 𝐴)
9 offval2.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝑋)
109ralrimiva 2440 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
11 eqid 2083 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1211fnmpt 5077 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1310, 12syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
14 offval2.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1514fneq1d 5041 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1613, 15mpbird 165 . . 3 (𝜑𝐺 Fn 𝐴)
17 offval2.1 . . 3 (𝜑𝐴𝑉)
18 inidm 3192 . . 3 (𝐴𝐴) = 𝐴
196adantr 270 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2019fveq1d 5233 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2114adantr 270 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2221fveq1d 5233 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
238, 16, 17, 17, 18, 20, 22offval 5772 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
24 nffvmpt1 5239 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
25 nfcv 2223 . . . . 5 𝑥𝑅
26 nffvmpt1 5239 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
2724, 25, 26nfov 5588 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
28 nfcv 2223 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
29 fveq2 5231 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
30 fveq2 5231 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3129, 30oveq12d 5583 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3227, 28, 31cbvmpt 3893 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
33 simpr 108 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
343fvmpt2 5308 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 1, 34syl2anc 403 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3611fvmpt2 5308 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3733, 9, 36syl2anc 403 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3835, 37oveq12d 5583 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
3938mpteq2dva 3889 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4032, 39syl5eq 2127 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4123, 40eqtrd 2115 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wral 2353  cmpt 3860   Fn wfn 4948  cfv 4953  (class class class)co 5565  𝑓 cof 5763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-of 5765
This theorem is referenced by:  ofc12  5784  caofinvl  5786  caofcom  5787
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