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Theorem opabbrfexd 44740
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 pm4.24 564 . . 3 (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦𝑥𝑅𝑦))
21opabbii 5146 . 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 44739 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑥𝑅𝑦)} ∈ V)
92, 8eqeltrid 2845 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2110  Vcvv 3431   class class class wbr 5079  {copab 5141  wf 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439
This theorem is referenced by: (None)
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