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Theorem reldm 7110
 Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
reldm (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem reldm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 releldm2 7109 . . 3 (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
2 fvex 6113 . . . . . 6 (1st𝑥) ∈ V
3 eqid 2610 . . . . . 6 (𝑥𝐴 ↦ (1st𝑥)) = (𝑥𝐴 ↦ (1st𝑥))
42, 3fnmpti 5935 . . . . 5 (𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴
5 fvelrnb 6153 . . . . 5 ((𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦))
64, 5ax-mp 5 . . . 4 (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦)
7 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
8 fvex 6113 . . . . . . . 8 (1st𝑧) ∈ V
97, 3, 8fvmpt 6191 . . . . . . 7 (𝑧𝐴 → ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = (1st𝑧))
109eqeq1d 2612 . . . . . 6 (𝑧𝐴 → (((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ (1st𝑧) = 𝑦))
1110rexbiia 3022 . . . . 5 (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦)
1211a1i 11 . . . 4 (Rel 𝐴 → (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
136, 12syl5rbb 272 . . 3 (Rel 𝐴 → (∃𝑧𝐴 (1st𝑧) = 𝑦𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
141, 13bitrd 267 . 2 (Rel 𝐴 → (𝑦 ∈ dom 𝐴𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
1514eqrdv 2608 1 (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  Rel wrel 5043   Fn wfn 5799  ‘cfv 5804  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  fidomdm  8128  dmct  28877
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