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Theorem oprabco 6237
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 277 . . 3 ((𝐻 Fn 𝐷 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)
4 oprabco.2 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
54a1i 9 . . 3 (𝐻 Fn 𝐷𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
6 dffn5im 5578 . . 3 (𝐻 Fn 𝐷𝐻 = (𝑧𝐷 ↦ (𝐻𝑧)))
7 fveq2 5531 . . 3 (𝑧 = 𝐶 → (𝐻𝑧) = (𝐻𝐶))
83, 5, 6, 7fmpoco 6236 . 2 (𝐻 Fn 𝐷 → (𝐻𝐹) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶)))
91, 8eqtr4id 2241 1 (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  ccom 4645   Fn wfn 5227  cfv 5232  cmpo 5894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161
This theorem is referenced by:  oprab2co  6238
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