Theorem opabresexd 47258

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	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
opabresexd.y	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)
opabresexd.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)
opabresexd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
opabresexd.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
opabresexd	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ 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

Step	Hyp	Ref	Expression
1		opabresexd.x	. 2 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
2		opabresexd.y	. 2 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)
3		opabresexd.a	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)
4		opabresexd.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
5		mapex 7926	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → {𝑦 ∣ 𝑦:𝐴⟶𝐵} ∈ V)
6	3, 4, 5	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝑦:𝐴⟶𝐵} ∈ V)
7		opabresexd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
8	1, 2, 6, 7	opabresex0d 47256	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2708  Vcvv 3455   class class class wbr 5115  {copab 5177  ⟶wf 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-xp 5652  df-rel 5653  df-cnv 5654  df-dm 5656  df-rn 5657  df-fun 6521  df-fn 6522  df-f 6523

This theorem is referenced by:  opabbrfexd  47259