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Theorem op1steq 8043
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 5698 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3978 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 eqid 2728 . . . . . 6 (2nd𝐴) = (2nd𝐴)
4 eqopi 8035 . . . . . 6 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = (2nd𝐴))) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
53, 4mpanr2 702 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
6 fvex 6915 . . . . . 6 (2nd𝐴) ∈ V
7 opeq2 4879 . . . . . . 7 (𝑥 = (2nd𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd𝐴)⟩)
87eqeq2d 2739 . . . . . 6 (𝑥 = (2nd𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd𝐴)⟩))
96, 8spcev 3595 . . . . 5 (𝐴 = ⟨𝐵, (2nd𝐴)⟩ → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
105, 9syl 17 . . . 4 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)
1110ex 411 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
12 eqop 8041 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥)))
13 simpl 481 . . . . 5 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥) → (1st𝐴) = 𝐵)
1412, 13biimtrdi 252 . . . 4 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1514exlimdv 1928 . . 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 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  cop 4638   × cxp 5680  cfv 6553  1st c1st 7997  2nd c2nd 7998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 7999  df-2nd 8000
This theorem is referenced by:  releldm2  8053
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