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Theorem fnopabco 33997
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 474 . . 3 ((𝐻 Fn 𝐶𝑥𝐴) → 𝐵𝐶)
3 fnopabco.2 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-mpt 4921 . . . . 5 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtr4i 2822 . . . 4 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . 3 (𝐻 Fn 𝐶𝐹 = (𝑥𝐴𝐵))
7 dffn5 6464 . . . 4 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
87biimpi 208 . . 3 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
9 fveq2 6409 . . 3 (𝑦 = 𝐵 → (𝐻𝑦) = (𝐻𝐵))
102, 6, 8, 9fmptco 6621 . 2 (𝐻 Fn 𝐶 → (𝐻𝐹) = (𝑥𝐴 ↦ (𝐻𝐵)))
11 fnopabco.3 . . 3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
12 df-mpt 4921 . . 3 (𝑥𝐴 ↦ (𝐻𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
1311, 12eqtr4i 2822 . 2 𝐺 = (𝑥𝐴 ↦ (𝐻𝐵))
1410, 13syl6reqr 2850 1 (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {copab 4903  cmpt 4920  ccom 5314   Fn wfn 6094  cfv 6099
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
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-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
This theorem is referenced by:  opropabco  33998
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