Theorem oprabco 7936

Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Hypotheses

Ref	Expression
oprabco.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
oprabco.2	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
oprabco.3	⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))

Assertion

Ref	Expression
oprabco	⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem oprabco

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oprabco.3	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))
2		oprabco.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
3	2	adantl 482	. . 3 ⊢ ((𝐻 Fn 𝐷 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
4		oprabco.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
5	4	a1i 11	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
6		dffn5 6828	. . . 4 ⊢ (𝐻 Fn 𝐷 ↔ 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
7	6	biimpi 215	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
8		fveq2 6774	. . 3 ⊢ (𝑧 = 𝐶 → (𝐻‘𝑧) = (𝐻‘𝐶))
9	3, 5, 7, 8	fmpoco 7935	. 2 ⊢ (𝐻 Fn 𝐷 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)))
10	1, 9	eqtr4id 2797	1 ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157   ∘ ccom 5593   Fn wfn 6428  ‘cfv 6433   ∈ cmpo 7277

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  ax-un 7588

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-iun 4926  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  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  oprab2co  7937