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Theorem 1stfo 5505
 Description: 1st 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
1stfo 1st :V–onto→V

Proof of Theorem 1stfo
Dummy variables x y z w t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 5119 . . . 4 (Fun 1stxyz((x1st y x1st z) → y = z))
2 vex 2862 . . . . . . . . 9 y V
32br1st 4858 . . . . . . . 8 (x1st yw x = y, w)
4 vex 2862 . . . . . . . . 9 z V
54br1st 4858 . . . . . . . 8 (x1st zt x = z, t)
63, 5anbi12i 678 . . . . . . 7 ((x1st y x1st z) ↔ (w x = y, w t x = z, t))
7 eeanv 1913 . . . . . . 7 (wt(x = y, w x = z, t) ↔ (w x = y, w t x = z, t))
86, 7bitr4i 243 . . . . . 6 ((x1st y x1st z) ↔ wt(x = y, w x = z, t))
9 eqtr2 2371 . . . . . . . 8 ((x = y, w x = z, t) → y, w = z, t)
10 opth 4602 . . . . . . . . 9 (y, w = z, t ↔ (y = z w = t))
1110simplbi 446 . . . . . . . 8 (y, w = z, ty = z)
129, 11syl 15 . . . . . . 7 ((x = y, w x = z, t) → y = z)
1312exlimivv 1635 . . . . . 6 (wt(x = y, w x = z, t) → y = z)
148, 13sylbi 187 . . . . 5 ((x1st y x1st z) → y = z)
1514gen2 1547 . . . 4 yz((x1st y x1st z) → y = z)
161, 15mpgbir 1550 . . 3 Fun 1st
17 eqv 3565 . . . 4 (dom 1st = V ↔ x x dom 1st )
18 opeq 4619 . . . . 5 x = Proj1 x, Proj2 x
19 eqid 2353 . . . . . . 7 Proj1 x = Proj1 x
20 vex 2862 . . . . . . . . 9 x V
2120proj1ex 4593 . . . . . . . 8 Proj1 x V
2220proj2ex 4594 . . . . . . . 8 Proj2 x V
2321, 22opbr1st 5501 . . . . . . 7 ( Proj1 x, Proj2 x1st Proj1 x Proj1 x = Proj1 x)
2419, 23mpbir 200 . . . . . 6 Proj1 x, Proj2 x1st Proj1 x
25 breldm 4911 . . . . . 6 ( Proj1 x, Proj2 x1st Proj1 x Proj1 x, Proj2 x dom 1st )
2624, 25ax-mp 5 . . . . 5 Proj1 x, Proj2 x dom 1st
2718, 26eqeltri 2423 . . . 4 x dom 1st
2817, 27mpgbir 1550 . . 3 dom 1st = V
29 df-fn 4790 . . 3 (1st Fn V ↔ (Fun 1st dom 1st = V))
3016, 28, 29mpbir2an 886 . 2 1st Fn V
31 eqv 3565 . . 3 (ran 1st = V ↔ x x ran 1st )
32 eqid 2353 . . . . 5 x = x
3320, 20opbr1st 5501 . . . . 5 (x, x1st xx = x)
3432, 33mpbir 200 . . . 4 x, x1st x
35 brelrn 4960 . . . 4 (x, x1st xx ran 1st )
3634, 35ax-mp 5 . . 3 x ran 1st
3731, 36mpgbir 1550 . 2 ran 1st = V
38 df-fo 4793 . 2 (1st :V–onto→V ↔ (1st Fn V ran 1st = V))
3930, 37, 38mpbir2an 886 1 1st :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  1st c1st 4717  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –onto→wfo 4779 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-fo 4793 This theorem is referenced by:  opfv1st  5514  fundmen  6043  xpassen  6057
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