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Theorem br1stg 4730
 Description: The binary relationship over the 1st function. (Contributed by SF, 5-Jan-2015.)
Assertion
Ref Expression
br1stg ((A V B W) → (A, B1st CA = C))

Proof of Theorem br1stg
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4578 . . . 4 (z = Az, w = A, w)
21breq1d 4649 . . 3 (z = A → (z, w1st CA, w1st C))
3 eqeq1 2359 . . 3 (z = A → (z = CA = C))
42, 3bibi12d 312 . 2 (z = A → ((z, w1st Cz = C) ↔ (A, w1st CA = C)))
5 opeq2 4579 . . . 4 (w = BA, w = A, B)
65breq1d 4649 . . 3 (w = B → (A, w1st CA, B1st C))
76bibi1d 310 . 2 (w = B → ((A, w1st CA = C) ↔ (A, B1st CA = C)))
8 df-br 4640 . . 3 (z, w1st Cz, w, C 1st )
9 el1st 4729 . . 3 (z, w, C 1stxyz, w, C = x, y, x)
10 eqcom 2355 . . . . . 6 (z, w, C = x, y, xx, y, x = z, w, C)
11 opth 4602 . . . . . . 7 (x, y, x = z, w, C ↔ (x, y = z, w x = C))
12 opth 4602 . . . . . . . . 9 (x, y = z, w ↔ (x = z y = w))
1312anbi1i 676 . . . . . . . 8 ((x, y = z, w x = C) ↔ ((x = z y = w) x = C))
14 df-3an 936 . . . . . . . 8 ((x = z y = w x = C) ↔ ((x = z y = w) x = C))
1513, 14bitr4i 243 . . . . . . 7 ((x, y = z, w x = C) ↔ (x = z y = w x = C))
1611, 15bitri 240 . . . . . 6 (x, y, x = z, w, C ↔ (x = z y = w x = C))
1710, 16bitri 240 . . . . 5 (z, w, C = x, y, x ↔ (x = z y = w x = C))
18172exbii 1583 . . . 4 (xyz, w, C = x, y, xxy(x = z y = w x = C))
19 vex 2862 . . . . 5 z V
20 vex 2862 . . . . 5 w V
21 eqeq1 2359 . . . . 5 (x = z → (x = Cz = C))
22 biidd 228 . . . . 5 (y = w → (z = Cz = C))
2319, 20, 21, 22ceqsex2v 2896 . . . 4 (xy(x = z y = w x = C) ↔ z = C)
2418, 23bitri 240 . . 3 (xyz, w, C = x, y, xz = C)
258, 9, 243bitri 262 . 2 (z, w1st Cz = C)
264, 7, 25vtocl2g 2918 1 ((A V B W) → (A, B1st CA = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561   class class class wbr 4639  1st c1st 4717 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723 This theorem is referenced by: (None)
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