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Theorem opropabco 34880
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 5585 . . 3 ((𝐵𝑅𝐶𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 584 . 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 7148 . . . . . 6 (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
87eqeq2i 2831 . . . . 5 (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
98anbi2i 622 . . . 4 ((𝑥𝐴𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
109opabbii 5124 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
116, 10eqtri 2841 . 2 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
124, 5, 11fnopabco 34879 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cop 4563  {copab 5119   × cxp 5546  ccom 5552   Fn wfn 6343  cfv 6348  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148
This theorem is referenced by: (None)
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