Theorem op1steq 7980

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 5641	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3918	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		eqid 2737	. . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
4		eqopi 7972	. . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
5	3, 4	mpanr2 705	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
6		fvex 6848	. . . . . 6 ⊢ (2nd ‘𝐴) ∈ V
7		opeq2 4818	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd ‘𝐴)⟩)
8	7	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩))
9	6, 8	spcev 3549	. . . . 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 412	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12		eqop 7978	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥)))
13		simpl 482	. . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵)
14	12, 13	biimtrdi 253	. . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
15	14	exlimdv 1935	. . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
16	11, 15	impbid 212	. 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 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   × cxp 5623  ‘cfv 6493  1^st c1st 7934  2^nd c2nd 7935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-1st 7936  df-2nd 7937

This theorem is referenced by:  releldm2  7990