Theorem opropabco 37921

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 5661	. . 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 7361	. . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
8	7	eqeq2i 2749	. . . . 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 5165	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
11	6, 10	eqtri 2759	. 2 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
12	4, 5, 11	fnopabco 37920	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  {copab 5160   × cxp 5622   ∘ ccom 5628   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361

This theorem is referenced by: (None)