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Theorem releldm2 5753
Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
releldm2 (Rel A → (B dom Ax A (1stx) = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem releldm2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . . 3 (B dom AB V)
21anim2i 324 . 2 ((Rel A B dom A) → (Rel A B V))
3 id 19 . . . . 5 ((1stx) = B → (1stx) = B)
4 vex 2554 . . . . . 6 x V
5 1stexg 5736 . . . . . 6 (x V → (1stx) V)
64, 5ax-mp 7 . . . . 5 (1stx) V
73, 6syl6eqelr 2126 . . . 4 ((1stx) = BB V)
87rexlimivw 2423 . . 3 (x A (1stx) = BB V)
98anim2i 324 . 2 ((Rel A x A (1stx) = B) → (Rel A B V))
10 eldm2g 4474 . . . 4 (B V → (B dom AyB, y A))
1110adantl 262 . . 3 ((Rel A B V) → (B dom AyB, y A))
12 df-rel 4295 . . . . . . . . 9 (Rel AA ⊆ (V × V))
13 ssel 2933 . . . . . . . . 9 (A ⊆ (V × V) → (x Ax (V × V)))
1412, 13sylbi 114 . . . . . . . 8 (Rel A → (x Ax (V × V)))
1514imp 115 . . . . . . 7 ((Rel A x A) → x (V × V))
16 op1steq 5747 . . . . . . 7 (x (V × V) → ((1stx) = By x = ⟨B, y⟩))
1715, 16syl 14 . . . . . 6 ((Rel A x A) → ((1stx) = By x = ⟨B, y⟩))
1817rexbidva 2317 . . . . 5 (Rel A → (x A (1stx) = Bx A y x = ⟨B, y⟩))
1918adantr 261 . . . 4 ((Rel A B V) → (x A (1stx) = Bx A y x = ⟨B, y⟩))
20 rexcom4 2571 . . . . 5 (x A y x = ⟨B, y⟩ ↔ yx A x = ⟨B, y⟩)
21 risset 2346 . . . . . 6 (⟨B, y Ax A x = ⟨B, y⟩)
2221exbii 1493 . . . . 5 (yB, y Ayx A x = ⟨B, y⟩)
2320, 22bitr4i 176 . . . 4 (x A y x = ⟨B, y⟩ ↔ yB, y A)
2419, 23syl6bb 185 . . 3 ((Rel A B V) → (x A (1stx) = ByB, y A))
2511, 24bitr4d 180 . 2 ((Rel A B V) → (B dom Ax A (1stx) = B))
262, 9, 25pm5.21nd 824 1 (Rel A → (B dom Ax A (1stx) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551  wss 2911  cop 3370   × cxp 4286  dom cdm 4288  Rel wrel 4293  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:  reldm  5754
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