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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  –onto→wfo 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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