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Theorem releldm2 5734
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 2543 . . 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 2538 . . . . . 6 x V
5 1stexg 5717 . . . . . 6 (x V → (1stx) V)
64, 5ax-mp 7 . . . . 5 (1stx) V
73, 6syl6eqelr 2111 . . . 4 ((1stx) = BB V)
87rexlimivw 2407 . . 3 (x A (1stx) = BB V)
98anim2i 324 . 2 ((Rel A x A (1stx) = B) → (Rel A B V))
10 eldm2g 4458 . . . 4 (B V → (B dom AyB, y A))
1110adantl 262 . . 3 ((Rel A B V) → (B dom AyB, y A))
12 df-rel 4279 . . . . . . . . 9 (Rel AA ⊆ (V × V))
13 ssel 2916 . . . . . . . . 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 5728 . . . . . . 7 (x (V × V) → ((1stx) = By x = ⟨B, y⟩))
1715, 16syl 14 . . . . . 6 ((Rel A x A) → ((1stx) = By x = ⟨B, y⟩))
1817rexbidva 2301 . . . . 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 2554 . . . . 5 (x A y x = ⟨B, y⟩ ↔ yx A x = ⟨B, y⟩)
21 risset 2330 . . . . . 6 (⟨B, y Ax A x = ⟨B, y⟩)
2221exbii 1478 . . . . 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 815 1 (Rel A → (B dom Ax A (1stx) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wrex 2285  Vcvv 2535  wss 2894  cop 3353   × cxp 4270  dom cdm 4272  Rel wrel 4277  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:  reldm  5735
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