Theorem releldm2 7984

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 3458	. . 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 6844	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2842	. . . 4 ⊢ ((1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
6	5	rexlimivw 3130	. . 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 5845	. . . 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 5628	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
11		ssel 3924	. . . . . . . . 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 7974	. . . . . . 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 3155	. . . . 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 3260	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19		risset 3208	. . . . . 6 ⊢ (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
20	19	exbii 1849	. . . . 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 801	1 ⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437   ⊆ wss 3898  ⟨cop 4583   × cxp 5619  dom cdm 5621  Rel wrel 5626  ‘cfv 6489  1^st c1st 7928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fv 6497  df-1st 7930  df-2nd 7931

This theorem is referenced by:  reldm  7985  releldmdifi  7986  satffunlem1lem2  35519  satffunlem2lem2  35522