Theorem uzin2 11731

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 9903	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5528	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3	1, 2	ax-mp 5	. . 3 |- ZZ>= Fn ZZ
4		fvelrnb 5744	. . 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 5744	. . 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 3425	. . 3 |- ( ( ZZ>= ` x ) = A -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) = ( A i^i ( ZZ>= ` y ) ) )
9	8	eleq1d 2307	. 2 |- ( ( ZZ>= ` x ) = A -> ( ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= ) )
10		ineq2 3426	. . 3 |- ( ( ZZ>= ` y ) = B -> ( A i^i ( ZZ>= ` y ) ) = ( A i^i B ) )
11	10	eleq1d 2307	. 2 |- ( ( ZZ>= ` y ) = B -> ( ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i B ) e. ran ZZ>= ) )
12		uzin 9934	. . 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 9700	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> DECID x <_ y )
16	13, 14, 15	ifcldcd 3675	. . . 4 |- ( ( x e. ZZ /\ y e. ZZ ) -> if ( x <_ y , y , x ) e. ZZ )
17		fnfvelrn 5831	. . . 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 418	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ZZ>= ` if ( x <_ y , y , x ) ) e. ran ZZ>= )
19	12, 18	eqeltrd 2315	. 2 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= )
20	5, 7, 9, 11, 19	2gencl 2855	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 1402   e. wcel 2209   E.wrex 2529   i^i cin 3219   ifcif 3635   ~Pcpw 3685   class class class wbr 4125   ran crn 4770   Fn wfn 5367   -->wf 5368   ` cfv 5372   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by:  rexanuz  11732