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Theorem fnopabco 36232
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 5193 . . 3 (𝑥𝐴 ↦ (𝐻𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐻𝐵))}
31, 2eqtr4i 2764 . 2 𝐺 = (𝑥𝐴 ↦ (𝐻𝐵))
4 fnopabco.1 . . . 4 (𝑥𝐴𝐵𝐶)
54adantl 483 . . 3 ((𝐻 Fn 𝐶𝑥𝐴) → 𝐵𝐶)
6 fnopabco.2 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
7 df-mpt 5193 . . . . 5 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
86, 7eqtr4i 2764 . . . 4 𝐹 = (𝑥𝐴𝐵)
98a1i 11 . . 3 (𝐻 Fn 𝐶𝐹 = (𝑥𝐴𝐵))
10 dffn5 6905 . . . 4 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
1110biimpi 215 . . 3 (𝐻 Fn 𝐶𝐻 = (𝑦𝐶 ↦ (𝐻𝑦)))
12 fveq2 6846 . . 3 (𝑦 = 𝐵 → (𝐻𝑦) = (𝐻𝐵))
135, 9, 11, 12fmptco 7079 . 2 (𝐻 Fn 𝐶 → (𝐻𝐹) = (𝑥𝐴 ↦ (𝐻𝐵)))
143, 13eqtr4id 2792 1 (𝐻 Fn 𝐶𝐺 = (𝐻𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {copab 5171  cmpt 5192  ccom 5641   Fn wfn 6495  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by:  opropabco  36233
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