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Theorem op1steq 6029
 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 4605 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3057 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 eqid 2113 . . . . . 6 (2nd𝐴) = (2nd𝐴)
4 eqopi 6022 . . . . . 6 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = (2nd𝐴))) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
53, 4mpanr2 432 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
6 2ndexg 6018 . . . . . . 7 (𝐴 ∈ (V × V) → (2nd𝐴) ∈ V)
7 opeq2 3670 . . . . . . . . 9 (𝑥 = (2nd𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd𝐴)⟩)
87eqeq2d 2124 . . . . . . . 8 (𝑥 = (2nd𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd𝐴)⟩))
98spcegv 2743 . . . . . . 7 ((2nd𝐴) ∈ V → (𝐴 = ⟨𝐵, (2nd𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
106, 9syl 14 . . . . . 6 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, (2nd𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
1110adantr 272 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → (𝐴 = ⟨𝐵, (2nd𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
125, 11mpd 13 . . . 4 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
1312ex 114 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
14 eqop 6027 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥)))
15 simpl 108 . . . . 5 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥) → (1st𝐴) = 𝐵)
1614, 15syl6bi 162 . . . 4 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1716exlimdv 1771 . . 3 (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1813, 17impbid 128 . 2 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
192, 18syl 14 1 (𝐴 ∈ (𝑉 × 𝑊) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312  ∃wex 1449   ∈ wcel 1461  Vcvv 2655  ⟨cop 3494   × cxp 4495  ‘cfv 5079  1st c1st 5988  2nd c2nd 5989 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fo 5085  df-fv 5087  df-1st 5990  df-2nd 5991 This theorem is referenced by:  releldm2  6035
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