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Theorem opelresg 5539
 Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.) (Revised by BJ, 18-Feb-2022.)
Assertion
Ref Expression
opelresg (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))

Proof of Theorem opelresg
StepHypRef Expression
1 df-res 5262 . . . 4 (𝐶𝐷) = (𝐶 ∩ (𝐷 × V))
21eleq2i 2842 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)))
32a1i 11 . 2 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V))))
4 elin 3947 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)))
54a1i 11 . 2 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V))))
6 opelxp 5285 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ (𝐴𝐷𝐵 ∈ V))
7 elex 3364 . . . . 5 (𝐵𝑉𝐵 ∈ V)
8 biid 251 . . . . . 6 ((𝐴𝐷𝐵 ∈ V) ↔ (𝐴𝐷𝐵 ∈ V))
98rbaib 528 . . . . 5 (𝐵 ∈ V → ((𝐴𝐷𝐵 ∈ V) ↔ 𝐴𝐷))
107, 9syl 17 . . . 4 (𝐵𝑉 → ((𝐴𝐷𝐵 ∈ V) ↔ 𝐴𝐷))
116, 10syl5bb 272 . . 3 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ 𝐴𝐷))
1211anbi2d 614 . 2 (𝐵𝑉 → ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
133, 5, 123bitrd 294 1 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722  ⟨cop 4323   × cxp 5248   ↾ cres 5252 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-res 5262 This theorem is referenced by:  brresg  5540  opelres  5541  opelresi  5548  issref  5649  setsnidel  41872
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