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Theorem offval2f 7056
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 397 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑊))
41, 3ralrimi 3106 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
5 offval2f.a . . . . . 6 𝑥𝐴
65fnmptf 6156 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
74, 6syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
8 offval2f.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
98fneq1d 6121 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
107, 9mpbird 247 . . 3 (𝜑𝐹 Fn 𝐴)
11 offval2f.3 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝑋)
1211ex 397 . . . . . 6 (𝜑 → (𝑥𝐴𝐶𝑋))
131, 12ralrimi 3106 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
145fnmptf 6156 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1513, 14syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
16 offval2f.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1716fneq1d 6121 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1815, 17mpbird 247 . . 3 (𝜑𝐺 Fn 𝐴)
19 offval2f.1 . . 3 (𝜑𝐴𝑉)
20 inidm 3971 . . 3 (𝐴𝐴) = 𝐴
218adantr 466 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2221fveq1d 6334 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2316adantr 466 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2423fveq1d 6334 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
2510, 18, 19, 19, 20, 22, 24offval 7051 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
26 nfcv 2913 . . . 4 𝑦𝐴
27 nffvmpt1 6340 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
28 nfcv 2913 . . . . 5 𝑥𝑅
29 nffvmpt1 6340 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
3027, 28, 29nfov 6821 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
31 nfcv 2913 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
32 fveq2 6332 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
33 fveq2 6332 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3432, 33oveq12d 6811 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3526, 5, 30, 31, 34cbvmptf 4882 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
36 simpr 471 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
375fvmpt2f 6425 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3836, 2, 37syl2anc 573 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
395fvmpt2f 6425 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4036, 11, 39syl2anc 573 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4138, 40oveq12d 6811 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
421, 41mpteq2da 4877 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4335, 42syl5eq 2817 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4425, 43eqtrd 2805 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wnf 1856  wcel 2145  wnfc 2900  wral 3061  cmpt 4863   Fn wfn 6026  cfv 6031  (class class class)co 6793  𝑓 cof 7042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044
This theorem is referenced by:  esumaddf  30463  binomcxplemnotnn0  39081
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