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Theorem opabbrfexd 46573
Description: A collection of ordered pairs, the second component being a function, is a set. (Contributed 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
opabbrfexd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabbrfexd
StepHypRef Expression
1 pm4.24 563 . . 3 (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦𝑥𝑅𝑦))
21opabbii 5208 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑥𝑅𝑦)}
3 opabresexd.x . . 3 ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
4 opabresexd.y . . 3 ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)
5 opabresexd.a . . 3 ((𝜑𝑥𝐶) → 𝐴𝑈)
6 opabresexd.b . . 3 ((𝜑𝑥𝐶) → 𝐵𝑉)
7 opabresexd.c . . 3 (𝜑𝐶𝑊)
83, 4, 5, 6, 7opabresexd 46572 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑥𝑅𝑦)} ∈ V)
92, 8eqeltrid 2831 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3468   class class class wbr 5141  {copab 5203  wf 6533
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-fun 6539  df-fn 6540  df-f 6541
This theorem is referenced by: (None)
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