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Theorem reldm 5735
 Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
reldm (Rel A → dom A = ran (x A ↦ (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 5734 . . 3 (Rel A → (y dom Az A (1stz) = y))
2 vex 2538 . . . . . . 7 x V
3 1stexg 5717 . . . . . . 7 (x V → (1stx) V)
42, 3ax-mp 7 . . . . . 6 (1stx) V
5 eqid 2022 . . . . . 6 (x A ↦ (1stx)) = (x A ↦ (1stx))
64, 5fnmpti 4953 . . . . 5 (x A ↦ (1stx)) Fn A
7 fvelrnb 5146 . . . . 5 ((x A ↦ (1stx)) Fn A → (y ran (x A ↦ (1stx)) ↔ z A ((x A ↦ (1stx))‘z) = y))
86, 7ax-mp 7 . . . 4 (y ran (x A ↦ (1stx)) ↔ z A ((x A ↦ (1stx))‘z) = y)
9 fveq2 5103 . . . . . . . 8 (x = z → (1stx) = (1stz))
10 vex 2538 . . . . . . . . 9 z V
11 1stexg 5717 . . . . . . . . 9 (z V → (1stz) V)
1210, 11ax-mp 7 . . . . . . . 8 (1stz) V
139, 5, 12fvmpt 5174 . . . . . . 7 (z A → ((x A ↦ (1stx))‘z) = (1stz))
1413eqeq1d 2030 . . . . . 6 (z A → (((x A ↦ (1stx))‘z) = y ↔ (1stz) = y))
1514rexbiia 2317 . . . . 5 (z A ((x A ↦ (1stx))‘z) = yz A (1stz) = y)
1615a1i 9 . . . 4 (Rel A → (z A ((x A ↦ (1stx))‘z) = yz A (1stz) = y))
178, 16syl5rbb 182 . . 3 (Rel A → (z A (1stz) = yy ran (x A ↦ (1stx))))
181, 17bitrd 177 . 2 (Rel A → (y dom Ay ran (x A ↦ (1stx))))
1918eqrdv 2020 1 (Rel A → dom A = ran (x A ↦ (1stx)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  Vcvv 2535   ↦ cmpt 3792  dom cdm 4272  ran crn 4273  Rel wrel 4277   Fn wfn 4824  ‘cfv 4829  1st c1st 5688 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691 This theorem is referenced by: (None)
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