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Theorem fnopabco 35881
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 5158 . . 3 (𝑥𝐴 ↦ (𝐻𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
31, 2eqtr4i 2769 . 2 𝐺 = (𝑥𝐴 ↦ (𝐻𝐵))
4 fnopabco.1 . . . 4 (𝑥𝐴𝐵𝐶)
54adantl 482 . . 3 ((𝐻 Fn 𝐶𝑥𝐴) → 𝐵𝐶)
6 fnopabco.2 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
7 df-mpt 5158 . . . . 5 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
86, 7eqtr4i 2769 . . . 4 𝐹 = (𝑥𝐴𝐵)
98a1i 11 . . 3 (𝐻 Fn 𝐶𝐹 = (𝑥𝐴𝐵))
10 dffn5 6828 . . . 4 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
1110biimpi 215 . . 3 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
12 fveq2 6774 . . 3 (𝑦 = 𝐵 → (𝐻𝑦) = (𝐻𝐵))
135, 9, 11, 12fmptco 7001 . 2 (𝐻 Fn 𝐶 → (𝐻𝐹) = (𝑥𝐴 ↦ (𝐻𝐵)))
143, 13eqtr4id 2797 1 (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {copab 5136  cmpt 5157  ccom 5593   Fn wfn 6428  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441
This theorem is referenced by:  opropabco  35882
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