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Theorem offval2f 6862
Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
Hypotheses
Ref Expression
offval2f.0 𝑥𝜑
offval2f.a 𝑥𝐴
offval2f.1 (𝜑𝐴𝑉)
offval2f.2 ((𝜑𝑥𝐴) → 𝐵𝑊)
offval2f.3 ((𝜑𝑥𝐴) → 𝐶𝑋)
offval2f.4 (𝜑𝐹 = (𝑥𝐴𝐵))
offval2f.5 (𝜑𝐺 = (𝑥𝐴𝐶))
Assertion
Ref Expression
offval2f (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable group:   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem offval2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 offval2f.0 . . . . . 6 𝑥𝜑
2 offval2f.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑊)
32ex 450 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑊))
41, 3ralrimi 2951 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
5 offval2f.a . . . . . 6 𝑥𝐴
65fnmptf 5973 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
74, 6syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
8 offval2f.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
98fneq1d 5939 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
107, 9mpbird 247 . . 3 (𝜑𝐹 Fn 𝐴)
11 offval2f.3 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝑋)
1211ex 450 . . . . . 6 (𝜑 → (𝑥𝐴𝐶𝑋))
131, 12ralrimi 2951 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
145fnmptf 5973 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1513, 14syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
16 offval2f.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1716fneq1d 5939 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1815, 17mpbird 247 . . 3 (𝜑𝐺 Fn 𝐴)
19 offval2f.1 . . 3 (𝜑𝐴𝑉)
20 inidm 3800 . . 3 (𝐴𝐴) = 𝐴
218adantr 481 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2221fveq1d 6150 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2316adantr 481 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2423fveq1d 6150 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
2510, 18, 19, 19, 20, 22, 24offval 6857 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
26 nfcv 2761 . . . 4 𝑦𝐴
27 nffvmpt1 6156 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
28 nfcv 2761 . . . . 5 𝑥𝑅
29 nffvmpt1 6156 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
3027, 28, 29nfov 6630 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
31 nfcv 2761 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
32 fveq2 6148 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
33 fveq2 6148 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3432, 33oveq12d 6622 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3526, 5, 30, 31, 34cbvmptf 4708 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
36 simpr 477 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
375fvmpt2f 6240 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3836, 2, 37syl2anc 692 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
395fvmpt2f 6240 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4036, 11, 39syl2anc 692 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4138, 40oveq12d 6622 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
421, 41mpteq2da 4703 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4335, 42syl5eq 2667 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4425, 43eqtrd 2655 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wnfc 2748  wral 2907  cmpt 4673   Fn wfn 5842  cfv 5847  (class class class)co 6604  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850
This theorem is referenced by:  esumaddf  29904  binomcxplemnotnn0  38037
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