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Theorem offval2f 6807
 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 449 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑊))
41, 3ralrimi 2940 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
5 offval2f.a . . . . . 6 𝑥𝐴
65fnmptf 5929 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
74, 6syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
8 offval2f.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
98fneq1d 5895 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
107, 9mpbird 246 . . 3 (𝜑𝐹 Fn 𝐴)
11 offval2f.3 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝑋)
1211ex 449 . . . . . 6 (𝜑 → (𝑥𝐴𝐶𝑋))
131, 12ralrimi 2940 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
145fnmptf 5929 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1513, 14syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
16 offval2f.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1716fneq1d 5895 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1815, 17mpbird 246 . . 3 (𝜑𝐺 Fn 𝐴)
19 offval2f.1 . . 3 (𝜑𝐴𝑉)
20 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
218adantr 480 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2221fveq1d 6105 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2316adantr 480 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2423fveq1d 6105 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
2510, 18, 19, 19, 20, 22, 24offval 6802 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
26 nfcv 2751 . . . 4 𝑦𝐴
27 nffvmpt1 6111 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
28 nfcv 2751 . . . . 5 𝑥𝑅
29 nffvmpt1 6111 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
3027, 28, 29nfov 6575 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
31 nfcv 2751 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
32 fveq2 6103 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
33 fveq2 6103 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3432, 33oveq12d 6567 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3526, 5, 30, 31, 34cbvmptf 4676 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
36 simpr 476 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
375fvmpt2f 6192 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3836, 2, 37syl2anc 691 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
395fvmpt2f 6192 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4036, 11, 39syl2anc 691 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4138, 40oveq12d 6567 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
421, 41mpteq2da 4671 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4335, 42syl5eq 2656 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4425, 43eqtrd 2644 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by:  esumaddf  29450  binomcxplemnotnn0  37577
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