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Theorem ot2ndg 5699
Description: Extract the second member of an ordered triple. (See ot1stg 5698 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot2ndg ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘(1st ‘⟨A, B, 𝐶⟩)) = B)

Proof of Theorem ot2ndg
StepHypRef Expression
1 df-ot 3356 . . . . 5 A, B, 𝐶⟩ = ⟨⟨A, B⟩, 𝐶
21fveq2i 5102 . . . 4 (1st ‘⟨A, B, 𝐶⟩) = (1st ‘⟨⟨A, B⟩, 𝐶⟩)
3 opexg 3934 . . . . . 6 ((A 𝑉 B 𝑊) → ⟨A, B V)
4 op1stg 5696 . . . . . 6 ((⟨A, B V 𝐶 𝑋) → (1st ‘⟨⟨A, B⟩, 𝐶⟩) = ⟨A, B⟩)
53, 4sylan 267 . . . . 5 (((A 𝑉 B 𝑊) 𝐶 𝑋) → (1st ‘⟨⟨A, B⟩, 𝐶⟩) = ⟨A, B⟩)
653impa 1083 . . . 4 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘⟨⟨A, B⟩, 𝐶⟩) = ⟨A, B⟩)
72, 6syl5eq 2062 . . 3 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘⟨A, B, 𝐶⟩) = ⟨A, B⟩)
87fveq2d 5103 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘(1st ‘⟨A, B, 𝐶⟩)) = (2nd ‘⟨A, B⟩))
9 op2ndg 5697 . . 3 ((A 𝑉 B 𝑊) → (2nd ‘⟨A, B⟩) = B)
1093adant3 910 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘⟨A, B⟩) = B)
118, 10eqtrd 2050 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘(1st ‘⟨A, B, 𝐶⟩)) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 871   = wceq 1226   wcel 1370  Vcvv 2531  cop 3349  cotp 3350  cfv 4825  1st c1st 5684  2nd c2nd 5685
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-ot 3356  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fv 4833  df-1st 5686  df-2nd 5687
This theorem is referenced by: (None)
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