MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  releldm2 Structured version   Visualization version   GIF version

Theorem releldm2 8067
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 3499 . . 3 (𝐵 ∈ dom 𝐴𝐵 ∈ V)
21anim2i 617 . 2 ((Rel 𝐴𝐵 ∈ dom 𝐴) → (Rel 𝐴𝐵 ∈ V))
3 id 22 . . . . 5 ((1st𝑥) = 𝐵 → (1st𝑥) = 𝐵)
4 fvex 6920 . . . . 5 (1st𝑥) ∈ V
53, 4eqeltrrdi 2848 . . . 4 ((1st𝑥) = 𝐵𝐵 ∈ V)
65rexlimivw 3149 . . 3 (∃𝑥𝐴 (1st𝑥) = 𝐵𝐵 ∈ V)
76anim2i 617 . 2 ((Rel 𝐴 ∧ ∃𝑥𝐴 (1st𝑥) = 𝐵) → (Rel 𝐴𝐵 ∈ V))
8 eldm2g 5913 . . . 4 (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
98adantl 481 . . 3 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
10 df-rel 5696 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
11 ssel 3989 . . . . . . . . 9 (𝐴 ⊆ (V × V) → (𝑥𝐴𝑥 ∈ (V × V)))
1210, 11sylbi 217 . . . . . . . 8 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
1312imp 406 . . . . . . 7 ((Rel 𝐴𝑥𝐴) → 𝑥 ∈ (V × V))
14 op1steq 8057 . . . . . . 7 (𝑥 ∈ (V × V) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1513, 14syl 17 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1615rexbidva 3175 . . . . 5 (Rel 𝐴 → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1716adantr 480 . . . 4 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18 rexcom4 3286 . . . . 5 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19 risset 3231 . . . . . 6 (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2019exbii 1845 . . . . 5 (∃𝑦𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2118, 20bitr4i 278 . . . 4 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴)
2217, 21bitrdi 287 . . 3 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
239, 22bitr4d 282 . 2 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
242, 7, 23pm5.21nd 802 1 (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  cop 4637   × cxp 5687  dom cdm 5689  Rel wrel 5694  cfv 6563  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  reldm  8068  releldmdifi  8069  satffunlem1lem2  35388  satffunlem2lem2  35391
  Copyright terms: Public domain W3C validator