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Theorem uzin2 15236
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 12773 . . . 4 β„€β‰₯:β„€βŸΆπ’« β„€
2 ffn 6673 . . . 4 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ β„€β‰₯ Fn β„€)
31, 2ax-mp 5 . . 3 β„€β‰₯ Fn β„€
4 fvelrnb 6908 . . 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 6908 . . 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 4170 . . 3 ((β„€β‰₯β€˜π‘₯) = 𝐴 β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) = (𝐴 ∩ (β„€β‰₯β€˜π‘¦)))
98eleq1d 2823 . 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 4171 . . 3 ((β„€β‰₯β€˜π‘¦) = 𝐡 β†’ (𝐴 ∩ (β„€β‰₯β€˜π‘¦)) = (𝐴 ∩ 𝐡))
1110eleq1d 2823 . 2 ((β„€β‰₯β€˜π‘¦) = 𝐡 β†’ ((𝐴 ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯ ↔ (𝐴 ∩ 𝐡) ∈ ran β„€β‰₯))
12 uzin 12810 . . 3 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) = (β„€β‰₯β€˜if(π‘₯ ≀ 𝑦, 𝑦, π‘₯)))
13 ifcl 4536 . . . . 5 ((𝑦 ∈ β„€ ∧ π‘₯ ∈ β„€) β†’ if(π‘₯ ≀ 𝑦, 𝑦, π‘₯) ∈ β„€)
1413ancoms 460 . . . 4 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(π‘₯ ≀ 𝑦, 𝑦, π‘₯) ∈ β„€)
15 fnfvelrn 7036 . . . 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 588 . . 3 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ (β„€β‰₯β€˜if(π‘₯ ≀ 𝑦, 𝑦, π‘₯)) ∈ ran β„€β‰₯)
1712, 16eqeltrd 2838 . 2 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘₯) ∩ (β„€β‰₯β€˜π‘¦)) ∈ ran β„€β‰₯)
185, 7, 9, 11, 172gencl 3489 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   ∩ cin 3914  ifcif 4491  π’« cpw 4565   class class class wbr 5110  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501   ≀ cle 11197  β„€cz 12506  β„€β‰₯cuz 12770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-pre-lttri 11132  ax-pre-lttrn 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-neg 11395  df-z 12507  df-uz 12771
This theorem is referenced by:  rexanuz  15237  zfbas  23263  heibor1lem  36297
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