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Theorem op1steq 5728
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) = Bx 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 4373 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 2918 . 2 (A (𝑉 × 𝑊) → A (V × V))
3 eqid 2022 . . . . . 6 (2ndA) = (2ndA)
4 eqopi 5721 . . . . . 6 ((A (V × V) ((1stA) = B (2ndA) = (2ndA))) → A = ⟨B, (2ndA)⟩)
53, 4mpanr2 416 . . . . 5 ((A (V × V) (1stA) = B) → A = ⟨B, (2ndA)⟩)
6 2ndexg 5718 . . . . . . 7 (A (V × V) → (2ndA) V)
7 opeq2 3524 . . . . . . . . 9 (x = (2ndA) → ⟨B, x⟩ = ⟨B, (2ndA)⟩)
87eqeq2d 2033 . . . . . . . 8 (x = (2ndA) → (A = ⟨B, x⟩ ↔ A = ⟨B, (2ndA)⟩))
98spcegv 2618 . . . . . . 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) = Bx A = ⟨B, x⟩))
14 eqop 5726 . . . . 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 1682 . . 3 (A (V × V) → (x A = ⟨B, x⟩ → (1stA) = B))
1813, 17impbid 120 . 2 (A (V × V) → ((1stA) = Bx A = ⟨B, x⟩))
192, 18syl 14 1 (A (𝑉 × 𝑊) → ((1stA) = Bx A = ⟨B, x⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cop 3353   × cxp 4270  cfv 4829  1st c1st 5688  2nd c2nd 5689
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by:  releldm2  5734
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