Theorem op1steq 8030

Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
op1steq	⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem op1steq

Step	Hyp	Ref	Expression
1		xpss 5678	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3941	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		eqid 2769	. . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
4		eqopi 8022	. . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
5	3, 4	mpanr2 716	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
6		fvex 6895	. . . . . 6 ⊢ (2nd ‘𝐴) ∈ V
7		opeq2 4843	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd ‘𝐴)⟩)
8	7	eqeq2d 2780	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩))
9	6, 8	spcev 3574	. . . . 5 ⊢ (𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
10	5, 9	syl 18	. . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
11	10	ex 417	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12		eqop 8028	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥)))
13		simpl 487	. . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵)
14	12, 13	biimtrdi 256	. . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
15	14	exlimdv 1960	. . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
16	11, 15	impbid 215	. 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
17	2, 16	syl 18	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   × cxp 5660  ‘cfv 6537  1^st c1st 7984  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7986  df-2nd 7987

This theorem is referenced by:  releldm2  8040