Theorem releldm2 6261

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 2782	. . 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 2774	. . . . . 6 ⊢ 𝑥 ∈ V
5		1stexg 6243	. . . . . 6 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
7	3, 6	eqeltrrdi 2296	. . . 4 ⊢ ((1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
8	7	rexlimivw 2618	. . 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 4872	. . . 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 4680	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
13		ssel 3186	. . . . . . . . 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 6255	. . . . . . 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 2502	. . . . 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 2794	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21		risset 2533	. . . . . 6 ⊢ (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
22	21	exbii 1627	. . . . 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 1372  ∃wex 1514   ∈ wcel 2175  ∃wrex 2484  Vcvv 2771   ⊆ wss 3165  ⟨cop 3635   × cxp 4671  dom cdm 4673  Rel wrel 4678  ‘cfv 5268  1^st c1st 6214

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fo 5274  df-fv 5276  df-1st 6216  df-2nd 6217

This theorem is referenced by:  reldm  6262