Theorem releldm2 8055

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 3471	. . 3 ⊢ (𝐵 ∈ dom 𝐴 → 𝐵 ∈ V)
2	1	anim2i 629	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ dom 𝐴) → (Rel 𝐴 ∧ 𝐵 ∈ V))
3		id 23	. . . . 5 ⊢ ((1st ‘𝑥) = 𝐵 → (1st ‘𝑥) = 𝐵)
4		fvex 6899	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2869	. . . 4 ⊢ ((1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
6	5	rexlimivw 3159	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 → 𝐵 ∈ V)
7	6	anim2i 629	. 2 ⊢ ((Rel 𝐴 ∧ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵) → (Rel 𝐴 ∧ 𝐵 ∈ V))
8		eldm2g 5880	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
9	8	adantl 487	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
10		df-rel 5657	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
11		ssel 3902	. . . . . . . . 9 ⊢ (𝐴 ⊆ (V × V) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
12	10, 11	sylbi 220	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
13	12	imp 412	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (V × V))
14		op1steq 8046	. . . . . . 7 ⊢ (𝑥 ∈ (V × V) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
15	13, 14	syl 18	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) = 𝐵 ↔ ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
16	15	rexbidva 3184	. . . . 5 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
17	16	adantr 486	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩))
18		rexcom4 3289	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
19		risset 3237	. . . . . 6 ⊢ (⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
20	19	exbii 1880	. . . . 5 ⊢ (∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑥 = ⟨𝐵, 𝑦⟩)
21	18, 20	bitr4i 281	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑥 = ⟨𝐵, 𝑦⟩ ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴)
22	17, 21	bitrdi 290	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵 ↔ ∃𝑦⟨𝐵, 𝑦⟩ ∈ 𝐴))
23	9, 22	bitr4d 285	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ V) → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
24	2, 7, 23	pm5.21nd 814	1 ⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   × cxp 5648  dom cdm 5650  Rel wrel 5655  ‘cfv 6539  1^st c1st 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-1st 8002  df-2nd 8003

This theorem is used by:  reldm  8056  releldmdifi  8057  satffunlem1lem2  36255  satffunlem2lem2  36258