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Theorem oprabco 6082
Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Hypotheses
Ref Expression
oprabco.1 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
oprabco.2 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
oprabco.3 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))
Assertion
Ref Expression
oprabco (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem oprabco
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 oprabco.1 . . . 4 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
21adantl 275 . . 3 ((𝐻 Fn 𝐷 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
3 oprabco.2 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43a1i 9 . . 3 (𝐻 Fn 𝐷𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
5 dffn5im 5435 . . 3 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
6 fveq2 5389 . . 3 (𝑧 = 𝐶 → (𝐻𝑧) = (𝐻𝐶))
72, 4, 5, 6fmpoco 6081 . 2 (𝐻 Fn 𝐷 → (𝐻𝐹) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶)))
8 oprabco.3 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))
97, 8syl6reqr 2169 1 (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  ccom 4513   Fn wfn 5088  cfv 5093  cmpo 5744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007
This theorem is referenced by:  oprab2co  6083
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