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Theorem op1steq 8058
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) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem op1steq
StepHypRef Expression
1 xpss 5701 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3979 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 eqid 2737 . . . . . 6 (2nd𝐴) = (2nd𝐴)
4 eqopi 8050 . . . . . 6 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = (2nd𝐴))) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
53, 4mpanr2 704 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
6 fvex 6919 . . . . . 6 (2nd𝐴) ∈ V
7 opeq2 4874 . . . . . . 7 (𝑥 = (2nd𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd𝐴)⟩)
87eqeq2d 2748 . . . . . 6 (𝑥 = (2nd𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd𝐴)⟩))
96, 8spcev 3606 . . . . 5 (𝐴 = ⟨𝐵, (2nd𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
105, 9syl 17 . . . 4 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
1110ex 412 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12 eqop 8056 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥)))
13 simpl 482 . . . . 5 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥) → (1st𝐴) = 𝐵)
1412, 13biimtrdi 253 . . . 4 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1514exlimdv 1933 . . 3 (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1611, 15impbid 212 . 2 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
172, 16syl 17 1 (𝐴 ∈ (𝑉 × 𝑊) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  releldm2  8068
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