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Theorem 2ndfo 5506
Description: 2nd is a mapping from the universe onto the universe. (Contributed by SF, 12-Feb-2015.) (Revised by Scott Fenton, 17-Apr-2021.)
Assertion
Ref Expression
2ndfo 2nd :V–onto→V

Proof of Theorem 2ndfo
Dummy variables x y z w t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 5119 . . . 4 (Fun 2ndxyz((x2nd y x2nd z) → y = z))
2 vex 2862 . . . . . . . . 9 y V
32br2nd 4859 . . . . . . . 8 (x2nd yw x = w, y)
4 vex 2862 . . . . . . . . 9 z V
54br2nd 4859 . . . . . . . 8 (x2nd zt x = t, z)
63, 5anbi12i 678 . . . . . . 7 ((x2nd y x2nd z) ↔ (w x = w, y t x = t, z))
7 eeanv 1913 . . . . . . 7 (wt(x = w, y x = t, z) ↔ (w x = w, y t x = t, z))
86, 7bitr4i 243 . . . . . 6 ((x2nd y x2nd z) ↔ wt(x = w, y x = t, z))
9 eqtr2 2371 . . . . . . . 8 ((x = w, y x = t, z) → w, y = t, z)
10 opth 4602 . . . . . . . . 9 (w, y = t, z ↔ (w = t y = z))
1110simprbi 450 . . . . . . . 8 (w, y = t, zy = z)
129, 11syl 15 . . . . . . 7 ((x = w, y x = t, z) → y = z)
1312exlimivv 1635 . . . . . 6 (wt(x = w, y x = t, z) → y = z)
148, 13sylbi 187 . . . . 5 ((x2nd y x2nd z) → y = z)
1514gen2 1547 . . . 4 yz((x2nd y x2nd z) → y = z)
161, 15mpgbir 1550 . . 3 Fun 2nd
17 eqv 3565 . . . 4 (dom 2nd = V ↔ x x dom 2nd )
18 opeq 4619 . . . . 5 x = Proj1 x, Proj2 x
19 eqid 2353 . . . . . . 7 Proj2 x = Proj2 x
20 vex 2862 . . . . . . . . 9 x V
2120proj1ex 4593 . . . . . . . 8 Proj1 x V
2220proj2ex 4594 . . . . . . . 8 Proj2 x V
2321, 22opbr2nd 5502 . . . . . . 7 ( Proj1 x, Proj2 x2nd Proj2 x Proj2 x = Proj2 x)
2419, 23mpbir 200 . . . . . 6 Proj1 x, Proj2 x2nd Proj2 x
25 breldm 4911 . . . . . 6 ( Proj1 x, Proj2 x2nd Proj2 x Proj1 x, Proj2 x dom 2nd )
2624, 25ax-mp 5 . . . . 5 Proj1 x, Proj2 x dom 2nd
2718, 26eqeltri 2423 . . . 4 x dom 2nd
2817, 27mpgbir 1550 . . 3 dom 2nd = V
29 df-fn 4790 . . 3 (2nd Fn V ↔ (Fun 2nd dom 2nd = V))
3016, 28, 29mpbir2an 886 . 2 2nd Fn V
31 eqv 3565 . . 3 (ran 2nd = V ↔ x x ran 2nd )
32 equid 1676 . . . . 5 x = x
3320, 20opbr2nd 5502 . . . . 5 (x, x2nd xx = x)
3432, 33mpbir 200 . . . 4 x, x2nd x
35 brelrn 4960 . . . 4 (x, x2nd xx ran 2nd )
3634, 35ax-mp 5 . . 3 x ran 2nd
3731, 36mpgbir 1550 . 2 ran 2nd = V
38 df-fo 4793 . 2 (2nd :V–onto→V ↔ (2nd Fn V ran 2nd = V))
3930, 37, 38mpbir2an 886 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358  wal 1540  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564   class class class wbr 4639  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  ontowfo 4779  2nd c2nd 4783
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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fo 4793  df-2nd 4797
This theorem is referenced by:  opfv2nd  5515  xpassen  6057
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