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Theorem offval2f 7417
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 416 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑊))
41, 3ralrimi 3210 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
5 offval2f.a . . . . . 6 𝑥𝐴
65fnmptf 6475 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
74, 6syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
8 offval2f.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
98fneq1d 6436 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
107, 9mpbird 260 . . 3 (𝜑𝐹 Fn 𝐴)
11 offval2f.3 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝑋)
1211ex 416 . . . . . 6 (𝜑 → (𝑥𝐴𝐶𝑋))
131, 12ralrimi 3210 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
145fnmptf 6475 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1513, 14syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
16 offval2f.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1716fneq1d 6436 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1815, 17mpbird 260 . . 3 (𝜑𝐺 Fn 𝐴)
19 offval2f.1 . . 3 (𝜑𝐴𝑉)
20 inidm 4180 . . 3 (𝐴𝐴) = 𝐴
218adantr 484 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2221fveq1d 6665 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2316adantr 484 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2423fveq1d 6665 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
2510, 18, 19, 19, 20, 22, 24offval 7412 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))))
26 nfcv 2982 . . . 4 𝑦𝐴
27 nffvmpt1 6674 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
28 nfcv 2982 . . . . 5 𝑥𝑅
29 nffvmpt1 6674 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
3027, 28, 29nfov 7181 . . . 4 𝑥(((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))
31 nfcv 2982 . . . 4 𝑦(((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
32 fveq2 6663 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
33 fveq2 6663 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3432, 33oveq12d 7169 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)) = (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3526, 5, 30, 31, 34cbvmptf 5152 . . 3 (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
36 simpr 488 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
375fvmpt2f 6762 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3836, 2, 37syl2anc 587 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
395fvmpt2f 6762 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4036, 11, 39syl2anc 587 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
4138, 40oveq12d 7169 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)) = (𝐵𝑅𝐶))
421, 41mpteq2da 5147 . . 3 (𝜑 → (𝑥𝐴 ↦ (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4335, 42syl5eq 2871 . 2 (𝜑 → (𝑦𝐴 ↦ (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦))) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
4425, 43eqtrd 2859 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2115  wnfc 2962  wral 3133  cmpt 5133   Fn wfn 6340  cfv 6345  (class class class)co 7151  f cof 7403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7405
This theorem is referenced by:  esumaddf  31405  binomcxplemnotnn0  41008
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