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Theorem opropabco 34061
 Description: Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
Hypotheses
Ref Expression
opropabco.1 (𝑥𝐴𝐵𝑅)
opropabco.2 (𝑥𝐴𝐶𝑆)
opropabco.3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = ⟨𝐵, 𝐶⟩)}
opropabco.4 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))}
Assertion
Ref Expression
opropabco (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
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
StepHypRef Expression
1 opropabco.1 . . 3 (𝑥𝐴𝐵𝑅)
2 opropabco.2 . . 3 (𝑥𝐴𝐶𝑆)
3 opelxpi 5379 . . 3 ((𝐵𝑅𝐶𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 581 . 2 (𝑥𝐴 → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
5 opropabco.3 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = ⟨𝐵, 𝐶⟩)}
6 opropabco.4 . . 3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))}
7 df-ov 6908 . . . . . 6 (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
87eqeq2i 2837 . . . . 5 (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
98anbi2i 618 . . . 4 ((𝑥𝐴𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
109opabbii 4940 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
116, 10eqtri 2849 . 2 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
124, 5, 11fnopabco 34060 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4403  {copab 4935   × cxp 5340   ∘ ccom 5346   Fn wfn 6118  ‘cfv 6123  (class class class)co 6905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908 This theorem is referenced by: (None)
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