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Theorem oprabco 7496
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 474 . . 3 ((𝐻 Fn 𝐷 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
3 oprabco.2 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43a1i 11 . . 3 (𝐻 Fn 𝐷𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
5 dffn5 6464 . . . 4 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
65biimpi 208 . . 3 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
7 fveq2 6409 . . 3 (𝑧 = 𝐶 → (𝐻𝑧) = (𝐻𝐶))
82, 4, 6, 7fmpt2co 7495 . 2 (𝐻 Fn 𝐷 → (𝐻𝐹) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶)))
9 oprabco.3 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))
108, 9syl6reqr 2850 1 (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cmpt 4920  ccom 5314   Fn wfn 6094  cfv 6099  cmpt2 6878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400
This theorem is referenced by:  oprab2co  7497
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