Theorem 1stfo 5506

Description: 1^st 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.

Step	Hyp	Ref	Expression
1		dffun2 5120	. . . 4 ⊢ (Fun 1st ↔ ∀x∀y∀z((x1st y ∧ x1st z) → y = z))
2		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
3	2	br1st 4859	. . . . . . . 8 ⊢ (x1st y ↔ ∃w x = ⟨y, w⟩)
4		vex 2863	. . . . . . . . 9 ⊢ z ∈ V
5	4	br1st 4859	. . . . . . . 8 ⊢ (x1st z ↔ ∃t x = ⟨z, t⟩)
6	3, 5	anbi12i 678	. . . . . . 7 ⊢ ((x1st y ∧ x1st z) ↔ (∃w x = ⟨y, w⟩ ∧ ∃t x = ⟨z, t⟩))
7		eeanv 1913	. . . . . . 7 ⊢ (∃w∃t(x = ⟨y, w⟩ ∧ x = ⟨z, t⟩) ↔ (∃w x = ⟨y, w⟩ ∧ ∃t x = ⟨z, t⟩))
8	6, 7	bitr4i 243	. . . . . 6 ⊢ ((x1st y ∧ x1st z) ↔ ∃w∃t(x = ⟨y, w⟩ ∧ x = ⟨z, t⟩))
9		eqtr2 2371	. . . . . . . 8 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨z, t⟩) → ⟨y, w⟩ = ⟨z, t⟩)
10		opth 4603	. . . . . . . . 9 ⊢ (⟨y, w⟩ = ⟨z, t⟩ ↔ (y = z ∧ w = t))
11	10	simplbi 446	. . . . . . . 8 ⊢ (⟨y, w⟩ = ⟨z, t⟩ → y = z)
12	9, 11	syl 15	. . . . . . 7 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨z, t⟩) → y = z)
13	12	exlimivv 1635	. . . . . 6 ⊢ (∃w∃t(x = ⟨y, w⟩ ∧ x = ⟨z, t⟩) → y = z)
14	8, 13	sylbi 187	. . . . 5 ⊢ ((x1st y ∧ x1st z) → y = z)
15	14	gen2 1547	. . . 4 ⊢ ∀y∀z((x1st y ∧ x1st z) → y = z)
16	1, 15	mpgbir 1550	. . 3 ⊢ Fun 1st
17		eqv 3566	. . . 4 ⊢ (dom 1st = V ↔ ∀x x ∈ dom 1st )
18		opeq 4620	. . . . 5 ⊢ x = ⟨ Proj1 x, Proj2 x⟩
19		eqid 2353	. . . . . . 7 ⊢ Proj1 x = Proj1 x
20		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
21	20	proj1ex 4594	. . . . . . . 8 ⊢ Proj1 x ∈ V
22	20	proj2ex 4595	. . . . . . . 8 ⊢ Proj2 x ∈ V
23	21, 22	opbr1st 5502	. . . . . . 7 ⊢ (⟨ Proj1 x, Proj2 x⟩1st Proj1 x ↔ Proj1 x = Proj1 x)
24	19, 23	mpbir 200	. . . . . 6 ⊢ ⟨ Proj1 x, Proj2 x⟩1st Proj1 x
25		breldm 4912	. . . . . 6 ⊢ (⟨ Proj1 x, Proj2 x⟩1st Proj1 x → ⟨ Proj1 x, Proj2 x⟩ ∈ dom 1st )
26	24, 25	ax-mp 5	. . . . 5 ⊢ ⟨ Proj1 x, Proj2 x⟩ ∈ dom 1st
27	18, 26	eqeltri 2423	. . . 4 ⊢ x ∈ dom 1st
28	17, 27	mpgbir 1550	. . 3 ⊢ dom 1st = V
29		df-fn 4791	. . 3 ⊢ (1st Fn V ↔ (Fun 1st ∧ dom 1st = V))
30	16, 28, 29	mpbir2an 886	. 2 ⊢ 1st Fn V
31		eqv 3566	. . 3 ⊢ (ran 1st = V ↔ ∀x x ∈ ran 1st )
32		eqid 2353	. . . . 5 ⊢ x = x
33	20, 20	opbr1st 5502	. . . . 5 ⊢ (⟨x, x⟩1st x ↔ x = x)
34	32, 33	mpbir 200	. . . 4 ⊢ ⟨x, x⟩1st x
35		brelrn 4961	. . . 4 ⊢ (⟨x, x⟩1st x → x ∈ ran 1st )
36	34, 35	ax-mp 5	. . 3 ⊢ x ∈ ran 1st
37	31, 36	mpgbir 1550	. 2 ⊢ ran 1st = V
38		df-fo 4794	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
39	30, 37, 38	mpbir2an 886	1 ⊢ 1st :V–onto→V

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640  1^st c1st 4718  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –onto→wfo 4780

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fo 4794

This theorem is referenced by:  opfv1st  5515  fundmen  6044  xpassen  6058