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Theorem op1steq 8023
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 5692 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3978 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 eqid 2731 . . . . . 6 (2nd𝐴) = (2nd𝐴)
4 eqopi 8015 . . . . . 6 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = (2nd𝐴))) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
53, 4mpanr2 701 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
6 fvex 6904 . . . . . 6 (2nd𝐴) ∈ V
7 opeq2 4874 . . . . . . 7 (𝑥 = (2nd𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd𝐴)⟩)
87eqeq2d 2742 . . . . . 6 (𝑥 = (2nd𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd𝐴)⟩))
96, 8spcev 3596 . . . . 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 8021 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥)))
13 simpl 482 . . . . 5 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥) → (1st𝐴) = 𝐵)
1412, 13biimtrdi 252 . . . 4 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1514exlimdv 1935 . . 3 (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1611, 15impbid 211 . 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 205  wa 395   = wceq 1540  wex 1780  wcel 2105  Vcvv 3473  cop 4634   × cxp 5674  cfv 6543  1st c1st 7977  2nd c2nd 7978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980
This theorem is referenced by:  releldm2  8033
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