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Theorem releldm2 6133
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 2723 . . 3 (𝐵 ∈ dom 𝐴𝐵 ∈ V)
21anim2i 340 . 2 ((Rel 𝐴𝐵 ∈ dom 𝐴) → (Rel 𝐴𝐵 ∈ V))
3 id 19 . . . . 5 ((1st𝑥) = 𝐵 → (1st𝑥) = 𝐵)
4 vex 2715 . . . . . 6 𝑥 ∈ V
5 1stexg 6115 . . . . . 6 (𝑥 ∈ V → (1st𝑥) ∈ V)
64, 5ax-mp 5 . . . . 5 (1st𝑥) ∈ V
73, 6eqeltrrdi 2249 . . . 4 ((1st𝑥) = 𝐵𝐵 ∈ V)
87rexlimivw 2570 . . 3 (∃𝑥𝐴 (1st𝑥) = 𝐵𝐵 ∈ V)
98anim2i 340 . 2 ((Rel 𝐴 ∧ ∃𝑥𝐴 (1st𝑥) = 𝐵) → (Rel 𝐴𝐵 ∈ V))
10 eldm2g 4782 . . . 4 (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
1110adantl 275 . . 3 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
12 df-rel 4593 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
13 ssel 3122 . . . . . . . . 9 (𝐴 ⊆ (V × V) → (𝑥𝐴𝑥 ∈ (V × V)))
1412, 13sylbi 120 . . . . . . . 8 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
1514imp 123 . . . . . . 7 ((Rel 𝐴𝑥𝐴) → 𝑥 ∈ (V × V))
16 op1steq 6127 . . . . . . 7 (𝑥 ∈ (V × V) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1715, 16syl 14 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1817rexbidva 2454 . . . . 5 (Rel 𝐴 → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1918adantr 274 . . . 4 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
20 rexcom4 2735 . . . . 5 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21 risset 2485 . . . . . 6 (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2221exbii 1585 . . . . 5 (∃𝑦𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2320, 22bitr4i 186 . . . 4 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴)
2419, 23bitrdi 195 . . 3 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
2511, 24bitr4d 190 . 2 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
262, 9, 25pm5.21nd 902 1 (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wex 1472  wcel 2128  wrex 2436  Vcvv 2712  wss 3102  cop 3563   × cxp 4584  dom cdm 4586  Rel wrel 4591  cfv 5170  1st c1st 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176  df-fv 5178  df-1st 6088  df-2nd 6089
This theorem is referenced by:  reldm  6134
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