Theorem eldm2 2528

Description: Membership in a domain. Theorem 4 of [Suppes] p. 59.

Hypothesis

Ref	Expression
eldm.1	⊢ A ∈ V

Assertion

Ref	Expression
eldm2	⊢ (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B)

Distinct variable group(s):   y,A   y,B

Proof of Theorem eldm2

Step	Hyp	Ref	Expression
1		eldm.1	. . 3 ⊢ A ∈ V
2	1	eldm 2527	. 2 ⊢ (A ∈ dom B ↔ ∃y ABy)
3		df-br 2063	. . 3 ⊢ (ABy ↔ ⟨A, y⟩ ∈ B)
4	3	biex 733	. 2 ⊢ (∃y ABy ↔ ∃y⟨A, y⟩ ∈ B)
5	2, 4	bitr 151	1 ⊢ (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B)

Syntax hints:   ↔ wb 127  ∃wex 678   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  dom cdm 2410

This theorem is referenced by:  eldmg 2529  dmss 2530  opeldm 2534  dmun 2536  dmin 2537  dmuni 2538  reldm0 2550  dmcosseq 2572  dmres 2584  iss 2599  relssdr 2668  dmco2 2673  dffun7 2688  funssres 2698  fn0 2739  dmfco 2864  tfrlem8 2956  tfrlem9 2957  1st2val 3097  tz9.12lem3 3505

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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