Theorem opropabco 38091

Description: Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Hypotheses

Ref	Expression
opropabco.1	⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅)
opropabco.2	⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆)
opropabco.3	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝐵, 𝐶⟩)}
opropabco.4	⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))}

Assertion

Ref	Expression
opropabco	⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑦,𝐶   𝑥,𝑀,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

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

Proof of Theorem opropabco

Step	Hyp	Ref	Expression
1		opropabco.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅)
2		opropabco.2	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆)
3		opelxpi 5655	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
4	1, 2, 3	syl2anc 590	. 2 ⊢ (𝑥 ∈ 𝐴 → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
5		opropabco.3	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝐵, 𝐶⟩)}
6		opropabco.4	. . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))}
7		df-ov 7359	. . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
8	7	eqeq2i 2752	. . . . 5 ⊢ (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
9	8	anbi2i 629	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
10	9	opabbii 5139	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
11	6, 10	eqtri 2762	. 2 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
12	4, 5, 11	fnopabco 38090	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4561  {copab 5134   × cxp 5616   ∘ ccom 5622   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359

This theorem is referenced by: (None)