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Theorem ot1stg 5702
Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5702, ot2ndg 5703, ot3rdgg 5704.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot1stg ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘(1st ‘⟨A, B, 𝐶⟩)) = A)

Proof of Theorem ot1stg
StepHypRef Expression
1 df-ot 3360 . . . . 5 A, B, 𝐶⟩ = ⟨⟨A, B⟩, 𝐶
21fveq2i 5106 . . . 4 (1st ‘⟨A, B, 𝐶⟩) = (1st ‘⟨⟨A, B⟩, 𝐶⟩)
3 opexg 3938 . . . . . 6 ((A 𝑉 B 𝑊) → ⟨A, B V)
4 op1stg 5700 . . . . . 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 1085 . . . 4 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘⟨⟨A, B⟩, 𝐶⟩) = ⟨A, B⟩)
72, 6syl5eq 2066 . . 3 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘⟨A, B, 𝐶⟩) = ⟨A, B⟩)
87fveq2d 5107 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘(1st ‘⟨A, B, 𝐶⟩)) = (1st ‘⟨A, B⟩))
9 op1stg 5700 . . 3 ((A 𝑉 B 𝑊) → (1st ‘⟨A, B⟩) = A)
1093adant3 912 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘⟨A, B⟩) = A)
118, 10eqtrd 2054 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (1st ‘(1st ‘⟨A, B, 𝐶⟩)) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353  cotp 3354  cfv 4829  1st c1st 5688
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-ot 3360  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-fv 4837  df-1st 5690
This theorem is referenced by: (None)
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