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Theorem releldm2 5862
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 𝐴 ↔ ∃𝑥𝐴 (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 2618 . . 3 (𝐵 ∈ dom 𝐴𝐵 ∈ V)
21anim2i 334 . 2 ((Rel 𝐴𝐵 ∈ dom 𝐴) → (Rel 𝐴𝐵 ∈ V))
3 id 19 . . . . 5 ((1st𝑥) = 𝐵 → (1st𝑥) = 𝐵)
4 vex 2612 . . . . . 6 𝑥 ∈ V
5 1stexg 5845 . . . . . 6 (𝑥 ∈ V → (1st𝑥) ∈ V)
64, 5ax-mp 7 . . . . 5 (1st𝑥) ∈ V
73, 6syl6eqelr 2174 . . . 4 ((1st𝑥) = 𝐵𝐵 ∈ V)
87rexlimivw 2478 . . 3 (∃𝑥𝐴 (1st𝑥) = 𝐵𝐵 ∈ V)
98anim2i 334 . 2 ((Rel 𝐴 ∧ ∃𝑥𝐴 (1st𝑥) = 𝐵) → (Rel 𝐴𝐵 ∈ V))
10 eldm2g 4579 . . . 4 (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
1110adantl 271 . . 3 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
12 df-rel 4398 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
13 ssel 3002 . . . . . . . . 9 (𝐴 ⊆ (V × V) → (𝑥𝐴𝑥 ∈ (V × V)))
1412, 13sylbi 119 . . . . . . . 8 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
1514imp 122 . . . . . . 7 ((Rel 𝐴𝑥𝐴) → 𝑥 ∈ (V × V))
16 op1steq 5856 . . . . . . 7 (𝑥 ∈ (V × V) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1715, 16syl 14 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1817rexbidva 2370 . . . . 5 (Rel 𝐴 → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1918adantr 270 . . . 4 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
20 rexcom4 2630 . . . . 5 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21 risset 2399 . . . . . 6 (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2221exbii 1537 . . . . 5 (∃𝑦𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2320, 22bitr4i 185 . . . 4 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴)
2419, 23syl6bb 194 . . 3 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
2511, 24bitr4d 189 . 2 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
262, 9, 25pm5.21nd 859 1 (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wrex 2354  Vcvv 2609  wss 2982  cop 3419   × cxp 4389  dom cdm 4391  Rel wrel 4396  cfv 4952  1st c1st 5816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2611  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fo 4958  df-fv 4960  df-1st 5818  df-2nd 5819
This theorem is referenced by:  reldm  5863
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