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Theorem reldm 6050
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 6049 . . 3 (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
2 vex 2661 . . . . . . 7 𝑥 ∈ V
3 1stexg 6031 . . . . . . 7 (𝑥 ∈ V → (1st𝑥) ∈ V)
42, 3ax-mp 5 . . . . . 6 (1st𝑥) ∈ V
5 eqid 2115 . . . . . 6 (𝑥𝐴 ↦ (1st𝑥)) = (𝑥𝐴 ↦ (1st𝑥))
64, 5fnmpti 5219 . . . . 5 (𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴
7 fvelrnb 5435 . . . . 5 ((𝑥𝐴 ↦ (1st𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦))
86, 7ax-mp 5 . . . 4 (𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥)) ↔ ∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦)
9 fveq2 5387 . . . . . . . 8 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
10 vex 2661 . . . . . . . . 9 𝑧 ∈ V
11 1stexg 6031 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
1210, 11ax-mp 5 . . . . . . . 8 (1st𝑧) ∈ V
139, 5, 12fvmpt 5464 . . . . . . 7 (𝑧𝐴 → ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = (1st𝑧))
1413eqeq1d 2124 . . . . . 6 (𝑧𝐴 → (((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ (1st𝑧) = 𝑦))
1514rexbiia 2425 . . . . 5 (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦)
1615a1i 9 . . . 4 (Rel 𝐴 → (∃𝑧𝐴 ((𝑥𝐴 ↦ (1st𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧𝐴 (1st𝑧) = 𝑦))
178, 16syl5rbb 192 . . 3 (Rel 𝐴 → (∃𝑧𝐴 (1st𝑧) = 𝑦𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
181, 17bitrd 187 . 2 (Rel 𝐴 → (𝑦 ∈ dom 𝐴𝑦 ∈ ran (𝑥𝐴 ↦ (1st𝑥))))
1918eqrdv 2113 1 (Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463  wrex 2392  Vcvv 2658  cmpt 3957  dom cdm 4507  ran crn 4508  Rel wrel 4512   Fn wfn 5086  cfv 5091  1st c1st 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-1st 6004  df-2nd 6005
This theorem is referenced by: (None)
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