Theorem oprabco 6122

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 275	. . 3 ⊢ ((𝐻 Fn 𝐷 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
4		oprabco.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
5	4	a1i 9	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
6		dffn5im 5475	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
7		fveq2 5429	. . 3 ⊢ (𝑧 = 𝐶 → (𝐻‘𝑧) = (𝐻‘𝐶))
8	3, 5, 6, 7	fmpoco 6121	. 2 ⊢ (𝐻 Fn 𝐷 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)))
9	1, 8	eqtr4id 2192	1 ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   ∘ ccom 4551   Fn wfn 5126  ‘cfv 5131   ∈ cmpo 5784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047

This theorem is referenced by:  oprab2co  6123