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Theorem opropabco 37197
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 5715 . . 3 ((𝐵𝑅𝐶𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 583 . 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 7423 . . . . . 6 (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
87eqeq2i 2741 . . . . 5 (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
98anbi2i 622 . . . 4 ((𝑥𝐴𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
109opabbii 5215 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
116, 10eqtri 2756 . 2 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
124, 5, 11fnopabco 37196 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cop 4635  {copab 5210   × cxp 5676  ccom 5682   Fn wfn 6543  cfv 6548  (class class class)co 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423
This theorem is referenced by: (None)
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