Theorem uzin2 11697

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 9874	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5513	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3	1, 2	ax-mp 5	. . 3 |- ZZ>= Fn ZZ
4		fvelrnb 5729	. . 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 5729	. . 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 3419	. . 3 |- ( ( ZZ>= ` x ) = A -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) = ( A i^i ( ZZ>= ` y ) ) )
9	8	eleq1d 2303	. 2 |- ( ( ZZ>= ` x ) = A -> ( ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= ) )
10		ineq2 3420	. . 3 |- ( ( ZZ>= ` y ) = B -> ( A i^i ( ZZ>= ` y ) ) = ( A i^i B ) )
11	10	eleq1d 2303	. 2 |- ( ( ZZ>= ` y ) = B -> ( ( A i^i ( ZZ>= ` y ) ) e. ran ZZ>= <-> ( A i^i B ) e. ran ZZ>= ) )
12		uzin 9905	. . 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 9671	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> DECID x <_ y )
16	13, 14, 15	ifcldcd 3664	. . . 4 |- ( ( x e. ZZ /\ y e. ZZ ) -> if ( x <_ y , y , x ) e. ZZ )
17		fnfvelrn 5814	. . . 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 2311	. 2 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( ZZ>= ` x ) i^i ( ZZ>= ` y ) ) e. ran ZZ>= )
20	5, 7, 9, 11, 19	2gencl 2849	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 1398   e. wcel 2205   E.wrex 2523   i^i cin 3213   ifcif 3624   ~Pcpw 3674   class class class wbr 4114   ran crn 4755   Fn wfn 5352   -->wf 5353   ` cfv 5357   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872

This theorem is referenced by:  rexanuz  11698