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Theorem opabresex0d 41073
Description: A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
Hypotheses
Ref Expression
opabresex0d.x ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
opabresex0d.t ((𝜑𝑥𝑅𝑦) → 𝜃)
opabresex0d.y ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
opabresex0d.c (𝜑𝐶𝑊)
Assertion
Ref Expression
opabresex0d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Distinct variable groups:   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabresex0d
StepHypRef Expression
1 opabresex0d.x . . . . 5 ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
2 opabresex0d.t . . . . 5 ((𝜑𝑥𝑅𝑦) → 𝜃)
31, 2jca 554 . . . 4 ((𝜑𝑥𝑅𝑦) → (𝑥𝐶𝜃))
43ex 450 . . 3 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
54alrimivv 1855 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
6 opabresex0d.c . . . 4 (𝜑𝐶𝑊)
76elexd 3212 . . 3 (𝜑𝐶 ∈ V)
8 opabresex0d.y . . . 4 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
98elexd 3212 . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ V)
107, 9opabex3d 7142 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V)
11 opabbrex 6692 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
125, 10, 11syl2anc 693 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480  wcel 1989  {cab 2607  Vcvv 3198   class class class wbr 4651  {copab 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894
This theorem is referenced by:  opabbrfex0d  41074  opabresexd  41075
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