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Theorem opropabco 34882
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 5586 . . 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 2834 . . . . 5 (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
98anbi2i 622 . . . 4 ((𝑥𝐴𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
109opabbii 5125 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
116, 10eqtri 2844 . 2 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
124, 5, 11fnopabco 34881 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cop 4565  {copab 5120   × cxp 5547  ccom 5553   Fn wfn 6344  cfv 6349  (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 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7148
This theorem is referenced by: (None)
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