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Theorem reldm 5812
 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 5811 . . 3 (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
2 vex 2560 . . . . . . 7 𝑥 ∈ V
3 1stexg 5794 . . . . . . 7 (𝑥 ∈ V → (1st𝑥) ∈ V)
42, 3ax-mp 7 . . . . . 6 (1st𝑥) ∈ V
5 eqid 2040 . . . . . 6 (𝑥𝐴 ↦ (1st𝑥)) = (𝑥𝐴 ↦ (1st𝑥))
64, 5fnmpti 5027 . . . . 5 (𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴
7 fvelrnb 5221 . . . . 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 5178 . . . . . . . 8 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
10 vex 2560 . . . . . . . . 9 𝑧 ∈ V
11 1stexg 5794 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
1210, 11ax-mp 7 . . . . . . . 8 (1st𝑧) ∈ V
139, 5, 12fvmpt 5249 . . . . . . 7 (𝑧𝐴 → ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = (1st𝑧))
1413eqeq1d 2048 . . . . . 6 (𝑧𝐴 → (((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ (1st𝑧) = 𝑦))
1514rexbiia 2339 . . . . 5 (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦)
1615a1i 9 . . . 4 (Rel 𝐴 → (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
178, 16syl5rbb 182 . . 3 (Rel 𝐴 → (∃𝑧𝐴 (1st𝑧) = 𝑦𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
181, 17bitrd 177 . 2 (Rel 𝐴 → (𝑦 ∈ dom 𝐴𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
1918eqrdv 2038 1 (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  Vcvv 2557   ↦ cmpt 3818  dom cdm 4345  ran crn 4346  Rel wrel 4350   Fn wfn 4897  ‘cfv 4902  1st c1st 5765 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by: (None)
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