Theorem releldm2 6238

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 2771	. . 3 ⊢ (𝐵 ∈ dom 𝐴 → 𝐵 ∈ V)
2	1	anim2i 342	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ dom 𝐴) → (Rel 𝐴 ∧ 𝐵 ∈ V))
3		id 19	. . . . 5 ⊢ ((1st ‘𝑥) = 𝐵 → (1st ‘𝑥) = 𝐵)
4		vex 2763	. . . . . 6 ⊢ 𝑥 ∈ V
5		1stexg 6220	. . . . . 6 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
7	3, 6	eqeltrrdi 2285	. . . 4 ⊢ ((1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
8	7	rexlimivw 2607	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
9	8	anim2i 342	. 2 ⊢ ((Rel 𝐴 ∧ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵) → (Rel 𝐴 ∧ 𝐵 ∈ V))
10		eldm2g 4858	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
11	10	adantl 277	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
12		df-rel 4666	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
13		ssel 3173	. . . . . . . . 9 ⊢ (𝐴 ⊆ (V × V) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
14	12, 13	sylbi 121	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
15	14	imp 124	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (V × V))
16		op1steq 6232	. . . . . . 7 ⊢ (𝑥 ∈ (V × V) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
17	15, 16	syl 14	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18	17	rexbidva 2491	. . . . 5 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
19	18	adantr 276	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
20		rexcom4 2783	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21		risset 2522	. . . . . 6 ⊢ (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
22	21	exbii 1616	. . . . 5 ⊢ (∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
23	20, 22	bitr4i 187	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴)
24	19, 23	bitrdi 196	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
25	11, 24	bitr4d 191	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
26	2, 9, 25	pm5.21nd 917	1 ⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760   ⊆ wss 3153  ⟨cop 3621   × cxp 4657  dom cdm 4659  Rel wrel 4664  ‘cfv 5254  1^st c1st 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-1st 6193  df-2nd 6194

This theorem is referenced by:  reldm  6239