Theorem fnopabco 37683

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 5250	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))}
3	1, 2	eqtr4i 2771	. 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 5250	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
8	6, 7	eqtr4i 2771	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	a1i 11	. . 3 ⊢ (𝐻 Fn 𝐶 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
10		dffn5 6980	. . . 4 ⊢ (𝐻 Fn 𝐶 ↔ 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦)))
11	10	biimpi 216	. . 3 ⊢ (𝐻 Fn 𝐶 → 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦)))
12		fveq2 6920	. . 3 ⊢ (𝑦 = 𝐵 → (𝐻‘𝑦) = (𝐻‘𝐵))
13	5, 9, 11, 12	fmptco 7163	. 2 ⊢ (𝐻 Fn 𝐶 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)))
14	3, 13	eqtr4id 2799	1 ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {copab 5228   ↦ cmpt 5249   ∘ ccom 5704   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  opropabco  37684