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Theorem op1steq 5805
 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 4446 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 2941 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 eqid 2040 . . . . . 6 (2nd𝐴) = (2nd𝐴)
4 eqopi 5798 . . . . . 6 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = (2nd𝐴))) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
53, 4mpanr2 414 . . . . 5 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = 𝐵) → 𝐴 = ⟨𝐵, (2nd𝐴)⟩)
6 2ndexg 5795 . . . . . . 7 (𝐴 ∈ (V × V) → (2nd𝐴) ∈ V)
7 opeq2 3550 . . . . . . . . 9 (𝑥 = (2nd𝐴) → ⟨𝐵, 𝑥⟩ = ⟨𝐵, (2nd𝐴)⟩)
87eqeq2d 2051 . . . . . . . 8 (𝑥 = (2nd𝐴) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝐵, (2nd𝐴)⟩))
98spcegv 2641 . . . . . . 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 261 . . . . 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 108 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = 𝐵 → ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
14 eqop 5803 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥)))
15 simpl 102 . . . . 5 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝑥) → (1st𝐴) = 𝐵)
1614, 15syl6bi 152 . . . 4 (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1716exlimdv 1700 . . 3 (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩ → (1st𝐴) = 𝐵))
1813, 17impbid 120 . 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 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   × cxp 4343  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by:  releldm2  5811
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