Theorem op1steq 6234

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 4768	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3176	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		eqid 2193	. . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
4		eqopi 6227	. . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
5	3, 4	mpanr2 438	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩)
6		2ndexg 6223	. . . . . . 7 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) ∈ V)
7		opeq2 3806	. . . . . . . . 9 ⊢ (𝑥 = (2nd ‘𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd ‘𝐴)⟩)
8	7	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩))
9	8	spcegv 2849	. . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ V → (𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
10	6, 9	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
11	10	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → (𝐴 = ⟨𝐵, (2nd ‘𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12	5, 11	mpd 13	. . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
13	12	ex 115	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
14		eqop 6232	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥)))
15		simpl 109	. . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵)
16	14, 15	biimtrdi 163	. . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
17	16	exlimdv 1830	. . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st ‘𝐴) = 𝐵))
18	13, 17	impbid 129	. 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
19	2, 18	syl 14	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760  ⟨cop 3622   × cxp 4658  ‘cfv 5255  1^st c1st 6193  2^nd c2nd 6194

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by:  releldm2  6240