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Theorem fnopabco 35536
Description: Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
fnopabco.1 (𝑥𝐴𝐵𝐶)
fnopabco.2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
fnopabco.3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
Assertion
Ref Expression
fnopabco (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Distinct variable groups:   𝑥,𝐶,𝑦   𝑦,𝐵   𝑥,𝐻,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fnopabco
StepHypRef Expression
1 fnopabco.3 . . 3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
2 df-mpt 5121 . . 3 (𝑥𝐴 ↦ (𝐻𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
31, 2eqtr4i 2765 . 2 𝐺 = (𝑥𝐴 ↦ (𝐻𝐵))
4 fnopabco.1 . . . 4 (𝑥𝐴𝐵𝐶)
54adantl 485 . . 3 ((𝐻 Fn 𝐶𝑥𝐴) → 𝐵𝐶)
6 fnopabco.2 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
7 df-mpt 5121 . . . . 5 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
86, 7eqtr4i 2765 . . . 4 𝐹 = (𝑥𝐴𝐵)
98a1i 11 . . 3 (𝐻 Fn 𝐶𝐹 = (𝑥𝐴𝐵))
10 dffn5 6740 . . . 4 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
1110biimpi 219 . . 3 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
12 fveq2 6686 . . 3 (𝑦 = 𝐵 → (𝐻𝑦) = (𝐻𝐵))
135, 9, 11, 12fmptco 6913 . 2 (𝐻 Fn 𝐶 → (𝐻𝐹) = (𝑥𝐴 ↦ (𝐻𝐵)))
143, 13eqtr4id 2793 1 (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {copab 5102  cmpt 5120  ccom 5539   Fn wfn 6344  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357
This theorem is referenced by:  opropabco  35537
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