Theorem releldm2 8068

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 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem releldm2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3501	. . 3 ⊢ (𝐵 ∈ dom 𝐴 → 𝐵 ∈ V)
2	1	anim2i 617	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ dom 𝐴) → (Rel 𝐴 ∧ 𝐵 ∈ V))
3		id 22	. . . . 5 ⊢ ((1st ‘𝑥) = 𝐵 → (1st ‘𝑥) = 𝐵)
4		fvex 6919	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2850	. . . 4 ⊢ ((1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
6	5	rexlimivw 3151	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
7	6	anim2i 617	. 2 ⊢ ((Rel 𝐴 ∧ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵) → (Rel 𝐴 ∧ 𝐵 ∈ V))
8		eldm2g 5910	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
9	8	adantl 481	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
10		df-rel 5692	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
11		ssel 3977	. . . . . . . . 9 ⊢ (𝐴 ⊆ (V × V) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
12	10, 11	sylbi 217	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
13	12	imp 406	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (V × V))
14		op1steq 8058	. . . . . . 7 ⊢ (𝑥 ∈ (V × V) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
15	13, 14	syl 17	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
16	15	rexbidva 3177	. . . . 5 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
17	16	adantr 480	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18		rexcom4 3288	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19		risset 3233	. . . . . 6 ⊢ (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
20	19	exbii 1848	. . . . 5 ⊢ (∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21	18, 20	bitr4i 278	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴)
22	17, 21	bitrdi 287	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
23	9, 22	bitr4d 282	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
24	2, 7, 23	pm5.21nd 802	1 ⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ⟨cop 4632   × cxp 5683  dom cdm 5685  Rel wrel 5690  ‘cfv 6561  1^st c1st 8012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  reldm  8069  releldmdifi  8070  satffunlem1lem2  35408  satffunlem2lem2  35411