Theorem uzin2 11538

Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Assertion

Ref	Expression
uzin2	|- ( ( A e. ran ZZ>= /\ B e. ran ZZ>= ) -> ( A i^i B ) e. ran ZZ>= )

Proof of Theorem uzin2

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

Step	Hyp	Ref	Expression
1		uzf 9748	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5479	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3	1, 2	ax-mp 5	. . 3 |- ZZ>= Fn ZZ
4		fvelrnb 5689	. . 3 |- ( ZZ>= Fn ZZ -> ( A e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = A ) )
5	3, 4	ax-mp 5	. 2 |- ( A e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = A )
6		fvelrnb 5689	. . 3 |- ( ZZ>= Fn ZZ -> ( B e. ran ZZ>= <-> E. y e. ZZ ( ZZ>= ` y ) = B ) )
7	3, 6	ax-mp 5	. 2 |- ( B e. ran ZZ>= <-> E. y e. ZZ ( ZZ>= ` y ) = B )
8		ineq1 3399	. . 3 |- ( ( ZZ>= ` x ) = A -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) = ( A i^i ( ZZ>= ` y ) ) )
9	8	eleq1d 2298	. 2 |- ( ( ZZ>= ` x ) = A -> ( ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= ) )
10		ineq2 3400	. . 3 |- ( ( ZZ>= ` y ) = B -> ( A i^i ( ZZ>= ` y ) ) = ( A i^i B ) )
11	10	eleq1d 2298	. 2 |- ( ( ZZ>= ` y ) = B -> ( ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i B ) e. ran ZZ>= ) )
12		uzin 9779	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) = ( ZZ>= ` if ( x <_ y , y , x ) ) )
13		simpr 110	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> y e. ZZ )
14		simpl 109	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> x e. ZZ )
15		zdcle 9546	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> DECID x <_ y )
16	13, 14, 15	ifcldcd 3641	. . . 4 |- ( ( x e. ZZ /\ y e. ZZ ) -> if ( x <_ y , y , x ) e. ZZ )
17		fnfvelrn 5775	. . . 4 |- ( ( ZZ>= Fn ZZ /\ if ( x <_ y , y , x ) e. ZZ ) -> ( ZZ>= ` if ( x <_ y , y , x ) ) e. ran ZZ>= )
18	3, 16, 17	sylancr 414	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ZZ>= ` if ( x <_ y , y , x ) ) e. ran ZZ>= )
19	12, 18	eqeltrd 2306	. 2 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= )
20	5, 7, 9, 11, 19	2gencl 2834	1 |- ( ( A e. ran ZZ>= /\ B e. ran ZZ>= ) -> ( A i^i B ) e. ran ZZ>= )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   i^i cin 3197   ifcif 3603   ~Pcpw 3650   class class class wbr 4086   ran crn 4724   Fn wfn 5319   -->wf 5320   ` cfv 5324   <_ cle 8205   ZZcz 9469   ZZ>=cuz 9745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746

This theorem is referenced by:  rexanuz  11539