Theorem opropabco 35878

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 5627	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
4	1, 2, 3	syl2anc 584	. 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 7274	. . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
8	7	eqeq2i 2753	. . . . 5 ⊢ (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
9	8	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
10	9	opabbii 5146	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
11	6, 10	eqtri 2768	. 2 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
12	4, 5, 11	fnopabco 35877	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ⟨cop 4573  {copab 5141   × cxp 5588   ∘ ccom 5594   Fn wfn 6427  ‘cfv 6432  (class class class)co 7271

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274

This theorem is referenced by: (None)