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Theorem opropabco 37328
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 582 . 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 7422 . . . . . 6 (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩)
87eqeq2i 2738 . . . . 5 (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))
98anbi2i 621 . . . 4 ((𝑥𝐴𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)))
109opabbii 5216 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
116, 10eqtri 2753 . 2 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))}
124, 5, 11fnopabco 37327 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4636  {copab 5211   × cxp 5676  ccom 5682   Fn wfn 6544  cfv 6549  (class class class)co 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  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 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422
This theorem is referenced by: (None)
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