Theorem op1steq 7727

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 5566	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3963	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		eqid 2821	. . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
4		eqopi 7719	. . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
5	3, 4	mpanr2 702	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
6		fvex 6678	. . . . . 6 ⊢ (2nd ‘𝐴) ∈ V
7		opeq2 4798	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd ‘𝐴)⟩)
8	7	eqeq2d 2832	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩))
9	6, 8	spcev 3607	. . . . 5 ⊢ (𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
10	5, 9	syl 17	. . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
11	10	ex 415	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12		eqop 7725	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥)))
13		simpl 485	. . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵)
14	12, 13	syl6bi 255	. . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
15	14	exlimdv 1930	. . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
16	11, 15	impbid 214	. 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
17	2, 16	syl 17	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3495  ⟨cop 4567   × cxp 5548  ‘cfv 6350  1^st c1st 7681  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fv 6358  df-1st 7683  df-2nd 7684

This theorem is referenced by:  releldm2  7736