Theorem opropabco 37686

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 5737	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
4	1, 2, 3	syl2anc 583	. 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 7453	. . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
8	7	eqeq2i 2753	. . . . 5 ⊢ (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
9	8	anbi2i 622	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
10	9	opabbii 5233	. . 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 37685	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654  {copab 5228   × cxp 5698   ∘ ccom 5704   Fn wfn 6570  ‘cfv 6575  (class class class)co 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ov 7453

This theorem is referenced by: (None)