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Theorem opabresexd 40603
Description: A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
Hypotheses
Ref Expression
opabresexd.x ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
opabresexd.y ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)
opabresexd.a ((𝜑𝑥𝐶) → 𝐴𝑈)
opabresexd.b ((𝜑𝑥𝐶) → 𝐵𝑉)
opabresexd.c (𝜑𝐶𝑊)
Assertion
Ref Expression
opabresexd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabresexd
StepHypRef Expression
1 opabresexd.x . 2 ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
2 opabresexd.y . 2 ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)
3 opabresexd.a . . 3 ((𝜑𝑥𝐶) → 𝐴𝑈)
4 opabresexd.b . . 3 ((𝜑𝑥𝐶) → 𝐵𝑉)
5 mapex 7808 . . 3 ((𝐴𝑈𝐵𝑉) → {𝑦𝑦:𝐴𝐵} ∈ V)
63, 4, 5syl2anc 692 . 2 ((𝜑𝑥𝐶) → {𝑦𝑦:𝐴𝐵} ∈ V)
7 opabresexd.c . 2 (𝜑𝐶𝑊)
81, 2, 6, 7opabresex0d 40601 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  {cab 2607  Vcvv 3186   class class class wbr 4613  {copab 4672  wf 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  opabbrfexd  40604
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