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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-bndl 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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