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Theorem reldm 5839
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 5838 . . 3 (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
2 vex 2577 . . . . . . 7 𝑥 ∈ V
3 1stexg 5821 . . . . . . 7 (𝑥 ∈ V → (1st𝑥) ∈ V)
42, 3ax-mp 7 . . . . . 6 (1st𝑥) ∈ V
5 eqid 2056 . . . . . 6 (𝑥𝐴 ↦ (1st𝑥)) = (𝑥𝐴 ↦ (1st𝑥))
64, 5fnmpti 5054 . . . . 5 (𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴
7 fvelrnb 5248 . . . . 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 5205 . . . . . . . 8 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
10 vex 2577 . . . . . . . . 9 𝑧 ∈ V
11 1stexg 5821 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
1210, 11ax-mp 7 . . . . . . . 8 (1st𝑧) ∈ V
139, 5, 12fvmpt 5276 . . . . . . 7 (𝑧𝐴 → ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = (1st𝑧))
1413eqeq1d 2064 . . . . . 6 (𝑧𝐴 → (((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ (1st𝑧) = 𝑦))
1514rexbiia 2356 . . . . 5 (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦)
1615a1i 9 . . . 4 (Rel 𝐴 → (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
178, 16syl5rbb 186 . . 3 (Rel 𝐴 → (∃𝑧𝐴 (1st𝑧) = 𝑦𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
181, 17bitrd 181 . 2 (Rel 𝐴 → (𝑦 ∈ dom 𝐴𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
1918eqrdv 2054 1 (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  wrex 2324  Vcvv 2574  cmpt 3845  dom cdm 4372  ran crn 4373  Rel wrel 4377   Fn wfn 4924  cfv 4929  1st c1st 5792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-1st 5794  df-2nd 5795
This theorem is referenced by: (None)
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