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Theorem reldm 5754
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 5753 . . 3 (Rel A → (y dom Az A (1stz) = y))
2 vex 2554 . . . . . . 7 x V
3 1stexg 5736 . . . . . . 7 (x V → (1stx) V)
42, 3ax-mp 7 . . . . . 6 (1stx) V
5 eqid 2037 . . . . . 6 (x A ↦ (1stx)) = (x A ↦ (1stx))
64, 5fnmpti 4970 . . . . 5 (x A ↦ (1stx)) Fn A
7 fvelrnb 5164 . . . . 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 5121 . . . . . . . 8 (x = z → (1stx) = (1stz))
10 vex 2554 . . . . . . . . 9 z V
11 1stexg 5736 . . . . . . . . 9 (z V → (1stz) V)
1210, 11ax-mp 7 . . . . . . . 8 (1stz) V
139, 5, 12fvmpt 5192 . . . . . . 7 (z A → ((x A ↦ (1stx))‘z) = (1stz))
1413eqeq1d 2045 . . . . . 6 (z A → (((x A ↦ (1stx))‘z) = y ↔ (1stz) = y))
1514rexbiia 2333 . . . . 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 2035 1 (Rel A → dom A = ran (x A ↦ (1stx)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wrex 2301  Vcvv 2551  cmpt 3809  dom cdm 4288  ran crn 4289  Rel wrel 4293   Fn wfn 4840  cfv 4845  1st c1st 5707
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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