Theorem dmopabss 4490

Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Assertion

Ref	Expression
dmopabss	⊢ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ⊆ A

Distinct variable group:   x,y,A

Allowed substitution hints:   φ(x,y)

Proof of Theorem dmopabss

Step	Hyp	Ref	Expression
1		dmopab 4489	. 2 ⊢ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} = {x ∣ ∃y(x ∈ A ∧ φ)}
2		19.42v 1783	. . . 4 ⊢ (∃y(x ∈ A ∧ φ) ↔ (x ∈ A ∧ ∃yφ))
3	2	abbii 2150	. . 3 ⊢ {x ∣ ∃y(x ∈ A ∧ φ)} = {x ∣ (x ∈ A ∧ ∃yφ)}
4		ssab2 3018	. . 3 ⊢ {x ∣ (x ∈ A ∧ ∃yφ)} ⊆ A
5	3, 4	eqsstri 2969	. 2 ⊢ {x ∣ ∃y(x ∈ A ∧ φ)} ⊆ A
6	1, 5	eqsstri 2969	1 ⊢ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ⊆ A

Syntax hints:   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  {cab 2023   ⊆ wss 2911  {copab 3808  dom cdm 4288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-dm 4298

This theorem is referenced by:  opabex  5328