Theorem oprabco 7793

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.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
2	1	adantl 484	. . 3 ⊢ ((𝐻 Fn 𝐷 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
3		oprabco.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	a1i 11	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
5		dffn5 6726	. . . 4 ⊢ (𝐻 Fn 𝐷 ↔ 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
6	5	biimpi 218	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
7		fveq2 6672	. . 3 ⊢ (𝑧 = 𝐶 → (𝐻‘𝑧) = (𝐻‘𝐶))
8	2, 4, 6, 7	fmpoco 7792	. 2 ⊢ (𝐻 Fn 𝐷 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)))
9		oprabco.3	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))
10	8, 9	syl6reqr 2877	1 ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5148   ∘ ccom 5561   Fn wfn 6352  ‘cfv 6357   ∈ cmpo 7160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692

This theorem is referenced by:  oprab2co  7794