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Theorem uzin2 15295
Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
uzin2 ((𝐴 ∈ ran β„€β‰₯ ∧ 𝐡 ∈ ran β„€β‰₯) β†’ (𝐴 ∩ 𝐡) ∈ ran β„€β‰₯)

Proof of Theorem uzin2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzf 12829 . . . 4 β„€β‰₯:β„€βŸΆπ’« β„€
2 ffn 6716 . . . 4 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ β„€β‰₯ Fn β„€)
31, 2ax-mp 5 . . 3 β„€β‰₯ Fn β„€
4 fvelrnb 6951 . . 3 (β„€β‰₯ Fn β„€ β†’ (𝐴 ∈ ran β„€β‰₯ ↔ βˆƒπ‘₯ ∈ β„€ (β„€β‰₯β€˜π‘₯) = 𝐴))
53, 4ax-mp 5 . 2 (𝐴 ∈ ran β„€β‰₯ ↔ βˆƒπ‘₯ ∈ β„€ (β„€β‰₯β€˜π‘₯) = 𝐴)
6 fvelrnb 6951 . . 3 (β„€β‰₯ Fn β„€ β†’ (𝐡 ∈ ran β„€β‰₯ ↔ βˆƒπ‘¦ ∈ β„€ (β„€β‰₯β€˜π‘¦) = 𝐡))
73, 6ax-mp 5 . 2 (𝐡 ∈ ran β„€β‰₯ ↔ βˆƒπ‘¦ ∈ β„€ (β„€β‰₯β€˜π‘¦) = 𝐡)
8 ineq1 4204 . . 3 ((β„€β‰₯β€˜π‘₯) = 𝐴 β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) = (𝐴 ∩ (β„€β‰₯β€˜π‘¦)))
98eleq1d 2816 . 2 ((β„€β‰₯β€˜π‘₯) = 𝐴 β†’ (((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯ ↔ (𝐴 ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯))
10 ineq2 4205 . . 3 ((β„€β‰₯β€˜π‘¦) = 𝐡 β†’ (𝐴 ∩ (β„€β‰₯β€˜π‘¦)) = (𝐴 ∩ 𝐡))
1110eleq1d 2816 . 2 ((β„€β‰₯β€˜π‘¦) = 𝐡 β†’ ((𝐴 ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯ ↔ (𝐴 ∩ 𝐡) ∈ ran β„€β‰₯))
12 uzin 12866 . . 3 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) = (β„€β‰₯β€˜if(π‘₯ ≀ 𝑦, 𝑦, π‘₯)))
13 ifcl 4572 . . . . 5 ((𝑦 ∈ β„€ ∧ π‘₯ ∈ β„€) β†’ if(π‘₯ ≀ 𝑦, 𝑦, π‘₯) ∈ β„€)
1413ancoms 457 . . . 4 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(π‘₯ ≀ 𝑦, 𝑦, π‘₯) ∈ β„€)
15 fnfvelrn 7081 . . . 4 ((β„€β‰₯ Fn β„€ ∧ if(π‘₯ ≀ 𝑦, 𝑦, π‘₯) ∈ β„€) β†’ (β„€β‰₯β€˜if(π‘₯ ≀ 𝑦, 𝑦, π‘₯)) ∈ ran β„€β‰₯)
163, 14, 15sylancr 585 . . 3 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ (β„€β‰₯β€˜if(π‘₯ ≀ 𝑦, 𝑦, π‘₯)) ∈ ran β„€β‰₯)
1712, 16eqeltrd 2831 . 2 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯)
185, 7, 9, 11, 172gencl 3515 1 ((𝐴 ∈ ran β„€β‰₯ ∧ 𝐡 ∈ ran β„€β‰₯) β†’ (𝐴 ∩ 𝐡) ∈ ran β„€β‰₯)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068   ∩ cin 3946  ifcif 4527  π’« cpw 4601   class class class wbr 5147  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542   ≀ cle 11253  β„€cz 12562  β„€β‰₯cuz 12826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-neg 11451  df-z 12563  df-uz 12827
This theorem is referenced by:  rexanuz  15296  zfbas  23620  heibor1lem  36980
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