New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  opfv1st GIF version

Theorem opfv1st 5514
 Description: The value of the 1st function on an ordered pair. (Contributed by SF, 23-Feb-2015.)
Hypotheses
Ref Expression
opfv1st.1 A V
opfv1st.2 B V
Assertion
Ref Expression
opfv1st (1stA, B) = A

Proof of Theorem opfv1st
StepHypRef Expression
1 eqid 2353 . . 3 A = A
2 opfv1st.1 . . . 4 A V
3 opfv1st.2 . . . 4 B V
42, 3opbr1st 5501 . . 3 (A, B1st AA = A)
51, 4mpbir 200 . 2 A, B1st A
6 1stfo 5505 . . . 4 1st :V–onto→V
7 fofn 5271 . . . 4 (1st :V–onto→V → 1st Fn V)
86, 7ax-mp 5 . . 3 1st Fn V
92, 3opex 4588 . . 3 A, B V
10 fnbrfvb 5358 . . 3 ((1st Fn V A, B V) → ((1stA, B) = AA, B1st A))
118, 9, 10mp2an 653 . 2 ((1stA, B) = AA, B1st A)
125, 11mpbir 200 1 (1stA, B) = A
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   Fn wfn 4776  –onto→wfo 4779   ‘cfv 4781 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  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795 This theorem is referenced by:  1st2nd2  5516  op1std  5522
 Copyright terms: Public domain W3C validator