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Theorem reldm 5891
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 5890 . . 3 (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
2 vex 2615 . . . . . . 7 𝑥 ∈ V
3 1stexg 5873 . . . . . . 7 (𝑥 ∈ V → (1st𝑥) ∈ V)
42, 3ax-mp 7 . . . . . 6 (1st𝑥) ∈ V
5 eqid 2083 . . . . . 6 (𝑥𝐴 ↦ (1st𝑥)) = (𝑥𝐴 ↦ (1st𝑥))
64, 5fnmpti 5095 . . . . 5 (𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴
7 fvelrnb 5297 . . . . 5 ((𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦))
86, 7ax-mp 7 . . . 4 (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦)
9 fveq2 5253 . . . . . . . 8 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
10 vex 2615 . . . . . . . . 9 𝑧 ∈ V
11 1stexg 5873 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
1210, 11ax-mp 7 . . . . . . . 8 (1st𝑧) ∈ V
139, 5, 12fvmpt 5326 . . . . . . 7 (𝑧𝐴 → ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = (1st𝑧))
1413eqeq1d 2091 . . . . . 6 (𝑧𝐴 → (((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ (1st𝑧) = 𝑦))
1514rexbiia 2387 . . . . 5 (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦)
1615a1i 9 . . . 4 (Rel 𝐴 → (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
178, 16syl5rbb 191 . . 3 (Rel 𝐴 → (∃𝑧𝐴 (1st𝑧) = 𝑦𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
181, 17bitrd 186 . 2 (Rel 𝐴 → (𝑦 ∈ dom 𝐴𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
1918eqrdv 2081 1 (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wrex 2354  Vcvv 2612  cmpt 3865  dom cdm 4401  ran crn 4402  Rel wrel 4406   Fn wfn 4964  cfv 4969  1st c1st 5844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-1st 5846  df-2nd 5847
This theorem is referenced by: (None)
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