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Theorem op1steq 5747
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 4389 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 2935 . 2 (A (𝑉 × 𝑊) → A (V × V))
3 eqid 2037 . . . . . 6 (2ndA) = (2ndA)
4 eqopi 5740 . . . . . 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 5737 . . . . . . 7 (A (V × V) → (2ndA) V)
7 opeq2 3541 . . . . . . . . 9 (x = (2ndA) → ⟨B, x⟩ = ⟨B, (2ndA)⟩)
87eqeq2d 2048 . . . . . . . 8 (x = (2ndA) → (A = ⟨B, x⟩ ↔ A = ⟨B, (2ndA)⟩))
98spcegv 2635 . . . . . . 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 5745 . . . . 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 1697 . . 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 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  releldm2  5753
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