Theorem r111 8638

Description: The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)

Assertion

Ref	Expression
r111	|- R1 : On -1-1-> _V

Proof of Theorem r111

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1fnon 8630	. . 3 |- R1 Fn On
2		dffn2 6047	. . 3 |- ( R1 Fn On <-> R1 : On --> _V )
3	1, 2	mpbi 220	. 2 |- R1 : On --> _V
4		eloni 5733	. . . . 5 |- ( x e. On -> Ord x )
5		eloni 5733	. . . . 5 |- ( y e. On -> Ord y )
6		ordtri3or 5755	. . . . 5 |- ( ( Ord x /\ Ord y ) -> ( x e. y \/ x = y \/ y e. x ) )
7	4, 5, 6	syl2an 494	. . . 4 |- ( ( x e. On /\ y e. On ) -> ( x e. y \/ x = y \/ y e. x ) )
8		sdomirr 8097	. . . . . . . . 9 |- -. ( R1 ` y ) ~< ( R1 ` y )
9		r1sdom 8637	. . . . . . . . . 10 |- ( ( y e. On /\ x e. y ) -> ( R1 ` x ) ~< ( R1 ` y ) )
10		breq1 4656	. . . . . . . . . 10 |- ( ( R1 ` x ) = ( R1 ` y ) -> ( ( R1 ` x ) ~< ( R1 ` y ) <-> ( R1 ` y ) ~< ( R1 ` y ) ) )
11	9, 10	syl5ibcom 235	. . . . . . . . 9 |- ( ( y e. On /\ x e. y ) -> ( ( R1 ` x ) = ( R1 ` y ) -> ( R1 ` y ) ~< ( R1 ` y ) ) )
12	8, 11	mtoi 190	. . . . . . . 8 |- ( ( y e. On /\ x e. y ) -> -. ( R1 ` x ) = ( R1 ` y ) )
13	12	3adant1 1079	. . . . . . 7 |- ( ( x e. On /\ y e. On /\ x e. y ) -> -. ( R1 ` x ) = ( R1 ` y ) )
14	13	pm2.21d 118	. . . . . 6 |- ( ( x e. On /\ y e. On /\ x e. y ) -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) )
15	14	3expia 1267	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) ) )
16		ax-1 6	. . . . . 6 |- ( x = y -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) )
17	16	a1i 11	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) ) )
18		r1sdom 8637	. . . . . . . . . 10 |- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) )
19		breq2 4657	. . . . . . . . . 10 |- ( ( R1 ` x ) = ( R1 ` y ) -> ( ( R1 ` y ) ~< ( R1 ` x ) <-> ( R1 ` y ) ~< ( R1 ` y ) ) )
20	18, 19	syl5ibcom 235	. . . . . . . . 9 |- ( ( x e. On /\ y e. x ) -> ( ( R1 ` x ) = ( R1 ` y ) -> ( R1 ` y ) ~< ( R1 ` y ) ) )
21	8, 20	mtoi 190	. . . . . . . 8 |- ( ( x e. On /\ y e. x ) -> -. ( R1 ` x ) = ( R1 ` y ) )
22	21	3adant2 1080	. . . . . . 7 |- ( ( x e. On /\ y e. On /\ y e. x ) -> -. ( R1 ` x ) = ( R1 ` y ) )
23	22	pm2.21d 118	. . . . . 6 |- ( ( x e. On /\ y e. On /\ y e. x ) -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) )
24	23	3expia 1267	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) ) )
25	15, 17, 24	3jaod 1392	. . . 4 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) ) )
26	7, 25	mpd 15	. . 3 |- ( ( x e. On /\ y e. On ) -> ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) )
27	26	rgen2a 2977	. 2 |- A. x e. On A. y e. On ( ( R1 ` x ) = ( R1 ` y ) -> x = y )
28		dff13 6512	. 2 |- ( R1 : On -1-1-> _V <-> ( R1 : On --> _V /\ A. x e. On A. y e. On ( ( R1 ` x ) = ( R1 ` y ) -> x = y ) ) )
29	3, 27, 28	mpbir2an 955	1 |- R1 : On -1-1-> _V

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   Ord word 5722   Oncon0 5723   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   ~< csdm 7954   R1cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-r1 8627

This theorem is referenced by:  tskinf  9591  grothomex  9651  rankeq1o  32278  elhf  32281  hfninf  32293