Theorem oprabco 6321

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 277	. . 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 5642	. . 3 ⊢ (𝐻 Fn 𝐷 → 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧)))
7		fveq2 5594	. . 3 ⊢ (𝑧 = 𝐶 → (𝐻‘𝑧) = (𝐻‘𝐶))
8	3, 5, 6, 7	fmpoco 6320	. 2 ⊢ (𝐻 Fn 𝐷 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)))
9	1, 8	eqtr4id 2258	1 ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ∘ ccom 4692   Fn wfn 5280  ‘cfv 5285   ∈ cmpo 5964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245

This theorem is referenced by:  oprab2co  6322