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Theorem fnopabco 34990
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.1 . . . 4 (𝑥𝐴𝐵𝐶)
21adantl 484 . . 3 ((𝐻 Fn 𝐶𝑥𝐴) → 𝐵𝐶)
3 fnopabco.2 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-mpt 5138 . . . . 5 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtr4i 2845 . . . 4 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . 3 (𝐻 Fn 𝐶𝐹 = (𝑥𝐴𝐵))
7 dffn5 6717 . . . 4 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
87biimpi 218 . . 3 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
9 fveq2 6663 . . 3 (𝑦 = 𝐵 → (𝐻𝑦) = (𝐻𝐵))
102, 6, 8, 9fmptco 6884 . 2 (𝐻 Fn 𝐶 → (𝐻𝐹) = (𝑥𝐴 ↦ (𝐻𝐵)))
11 fnopabco.3 . . 3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
12 df-mpt 5138 . . 3 (𝑥𝐴 ↦ (𝐻𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
1311, 12eqtr4i 2845 . 2 𝐺 = (𝑥𝐴 ↦ (𝐻𝐵))
1410, 13syl6reqr 2873 1 (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  {copab 5119  cmpt 5137  ccom 5552   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  opropabco  34991
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