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Theorem releldm2 7377
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 3344 . . 3 (𝐵 ∈ dom 𝐴𝐵 ∈ V)
21anim2i 594 . 2 ((Rel 𝐴𝐵 ∈ dom 𝐴) → (Rel 𝐴𝐵 ∈ V))
3 id 22 . . . . 5 ((1st𝑥) = 𝐵 → (1st𝑥) = 𝐵)
4 fvex 6354 . . . . 5 (1st𝑥) ∈ V
53, 4syl6eqelr 2840 . . . 4 ((1st𝑥) = 𝐵𝐵 ∈ V)
65rexlimivw 3159 . . 3 (∃𝑥𝐴 (1st𝑥) = 𝐵𝐵 ∈ V)
76anim2i 594 . 2 ((Rel 𝐴 ∧ ∃𝑥𝐴 (1st𝑥) = 𝐵) → (Rel 𝐴𝐵 ∈ V))
8 eldm2g 5467 . . . 4 (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
98adantl 473 . . 3 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
10 df-rel 5265 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
11 ssel 3730 . . . . . . . . 9 (𝐴 ⊆ (V × V) → (𝑥𝐴𝑥 ∈ (V × V)))
1210, 11sylbi 207 . . . . . . . 8 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
1312imp 444 . . . . . . 7 ((Rel 𝐴𝑥𝐴) → 𝑥 ∈ (V × V))
14 op1steq 7369 . . . . . . 7 (𝑥 ∈ (V × V) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1513, 14syl 17 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ((1st𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1615rexbidva 3179 . . . . 5 (Rel 𝐴 → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
1716adantr 472 . . . 4 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18 rexcom4 3357 . . . . 5 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19 risset 3192 . . . . . 6 (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2019exbii 1915 . . . . 5 (∃𝑦𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑥𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
2118, 20bitr4i 267 . . . 4 (∃𝑥𝐴𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴)
2217, 21syl6bb 276 . . 3 ((Rel 𝐴𝐵 ∈ V) → (∃𝑥𝐴 (1st𝑥) = 𝐵 ↔ ∃𝑦𝐵, 𝑦⟩ ∈ 𝐴))
239, 22bitr4d 271 . 2 ((Rel 𝐴𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
242, 7, 23pm5.21nd 979 1 (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wex 1845  wcel 2131  wrex 3043  Vcvv 3332  wss 3707  cop 4319   × cxp 5256  dom cdm 5258  Rel wrel 5263  cfv 6041  1st c1st 7323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-1st 7325  df-2nd 7326
This theorem is referenced by:  reldm  7378
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