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Theorem releldm2 8043
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 3482 . . 3 (𝐵 ∈ dom 𝐴𝐵 ∈ V)
21anim2i 615 . 2 ((Rel 𝐴𝐵 ∈ dom 𝐴) → (Rel 𝐴𝐵 ∈ V))
3 id 22 . . . . 5 ((1st𝑥) = 𝐵 → (1st𝑥) = 𝐵)
4 fvex 6903 . . . . 5 (1st𝑥) ∈ V
53, 4eqeltrrdi 2834 . . . 4 ((1st𝑥) = 𝐵𝐵 ∈ V)
65rexlimivw 3141 . . 3 (∃𝑥𝐴 (1st𝑥) = 𝐵𝐵 ∈ V)
76anim2i 615 . 2 ((Rel 𝐴 ∧ ∃𝑥𝐴 (1st𝑥) = 𝐵) → (Rel 𝐴𝐵 ∈ V))
8 eldm2g 5894 . . . 4 (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
98adantl 480 . . 3 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
10 df-rel 5677 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
11 ssel 3965 . . . . . . . . 9 (𝐴 ⊆ (V × V) → (𝑥𝐴𝑥 ∈ (V × V)))
1210, 11sylbi 216 . . . . . . . 8 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
1312imp 405 . . . . . . 7 ((Rel 𝐴𝑥𝐴) → 𝑥 ∈ (V × V))
14 op1steq 8033 . . . . . . 7 (𝑥 ∈ (V × V) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1513, 14syl 17 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1615rexbidva 3167 . . . . 5 (Rel 𝐴 → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1716adantr 479 . . . 4 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18 rexcom4 3276 . . . . 5 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19 risset 3221 . . . . . 6 (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2019exbii 1842 . . . . 5 (∃𝑦𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2118, 20bitr4i 277 . . . 4 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴)
2217, 21bitrdi 286 . . 3 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
239, 22bitr4d 281 . 2 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
242, 7, 23pm5.21nd 800 1 (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wrex 3060  Vcvv 3463  wss 3939  cop 4628   × cxp 5668  dom cdm 5670  Rel wrel 5675  cfv 6541  1st c1st 7987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6493  df-fun 6543  df-fv 6549  df-1st 7989  df-2nd 7990
This theorem is referenced by:  reldm  8044  releldmdifi  8045  satffunlem1lem2  35042  satffunlem2lem2  35045
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