Theorem fnopabco 37974

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	⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)
fnopabco.2	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
fnopabco.3	⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))}

Assertion

Ref	Expression
fnopabco	⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹))

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

Step	Hyp	Ref	Expression
1		fnopabco.3	. . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))}
2		df-mpt 5182	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))}
3	1, 2	eqtr4i 2763	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵))
4		fnopabco.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)
5	4	adantl 481	. . 3 ⊢ ((𝐻 Fn 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
6		fnopabco.2	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7		df-mpt 5182	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
8	6, 7	eqtr4i 2763	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	a1i 11	. . 3 ⊢ (𝐻 Fn 𝐶 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
10		dffn5 6900	. . . 4 ⊢ (𝐻 Fn 𝐶 ↔ 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦)))
11	10	biimpi 216	. . 3 ⊢ (𝐻 Fn 𝐶 → 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦)))
12		fveq2 6842	. . 3 ⊢ (𝑦 = 𝐵 → (𝐻‘𝑦) = (𝐻‘𝐵))
13	5, 9, 11, 12	fmptco 7084	. 2 ⊢ (𝐻 Fn 𝐶 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)))
14	3, 13	eqtr4id 2791	1 ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {copab 5162   ↦ cmpt 5181   ∘ ccom 5636   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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by:  opropabco  37975