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Theorem oprabco 8105
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.3 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))
2 oprabco.1 . . . 4 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
32adantl 480 . . 3 ((𝐻 Fn 𝐷 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
4 oprabco.2 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
54a1i 11 . . 3 (𝐻 Fn 𝐷𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
6 dffn5 6960 . . . 4 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
76biimpi 215 . . 3 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
8 fveq2 6900 . . 3 (𝑧 = 𝐶 → (𝐻𝑧) = (𝐻𝐶))
93, 5, 7, 8fmpoco 8104 . 2 (𝐻 Fn 𝐷 → (𝐻𝐹) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶)))
101, 9eqtr4id 2786 1 (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5233  ccom 5684   Fn wfn 6546  cfv 6551  cmpo 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998
This theorem is referenced by:  oprab2co  8106
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