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Theorem imasless 13442
Description: The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
imasless.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasless.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasless.f  |-  ( ph  ->  F : V -onto-> B
)
imasless.r  |-  ( ph  ->  R  e.  Z )
imasless.l  |-  .<_  =  ( le `  U )
Assertion
Ref Expression
imasless  |-  ( ph  -> 
.<_  C_  ( B  X.  B ) )

Proof of Theorem imasless
StepHypRef Expression
1 imasless.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasless.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasless.f . . 3  |-  ( ph  ->  F : V -onto-> B
)
4 imasless.r . . 3  |-  ( ph  ->  R  e.  Z )
5 eqid 2283 . . 3  |-  ( le
`  R )  =  ( le `  R
)
6 imasless.l . . 3  |-  .<_  =  ( le `  U )
71, 2, 3, 4, 5, 6imasle 13425 . 2  |-  ( ph  -> 
.<_  =  ( ( F  o.  ( le `  R ) )  o.  `' F ) )
8 relco 5171 . . . 4  |-  Rel  (
( F  o.  ( le `  R ) )  o.  `' F )
9 relssdmrn 5193 . . . 4  |-  ( Rel  ( ( F  o.  ( le `  R ) )  o.  `' F
)  ->  ( ( F  o.  ( le `  R ) )  o.  `' F )  C_  ( dom  ( ( F  o.  ( le `  R ) )  o.  `' F
)  X.  ran  (
( F  o.  ( le `  R ) )  o.  `' F ) ) )
108, 9ax-mp 8 . . 3  |-  ( ( F  o.  ( le
`  R ) )  o.  `' F ) 
C_  ( dom  (
( F  o.  ( le `  R ) )  o.  `' F )  X.  ran  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
11 dmco 5181 . . . . 5  |-  dom  (
( F  o.  ( le `  R ) )  o.  `' F )  =  ( `' `' F " dom  ( F  o.  ( le `  R ) ) )
12 fof 5451 . . . . . . . . 9  |-  ( F : V -onto-> B  ->  F : V --> B )
13 frel 5392 . . . . . . . . 9  |-  ( F : V --> B  ->  Rel  F )
143, 12, 133syl 18 . . . . . . . 8  |-  ( ph  ->  Rel  F )
15 dfrel2 5124 . . . . . . . 8  |-  ( Rel 
F  <->  `' `' F  =  F
)
1614, 15sylib 188 . . . . . . 7  |-  ( ph  ->  `' `' F  =  F
)
1716imaeq1d 5011 . . . . . 6  |-  ( ph  ->  ( `' `' F " dom  ( F  o.  ( le `  R ) ) )  =  ( F " dom  ( F  o.  ( le `  R ) ) ) )
18 imassrn 5025 . . . . . . 7  |-  ( F
" dom  ( F  o.  ( le `  R
) ) )  C_  ran  F
19 forn 5454 . . . . . . . 8  |-  ( F : V -onto-> B  ->  ran  F  =  B )
203, 19syl 15 . . . . . . 7  |-  ( ph  ->  ran  F  =  B )
2118, 20syl5sseq 3226 . . . . . 6  |-  ( ph  ->  ( F " dom  ( F  o.  ( le `  R ) ) )  C_  B )
2217, 21eqsstrd 3212 . . . . 5  |-  ( ph  ->  ( `' `' F " dom  ( F  o.  ( le `  R ) ) )  C_  B
)
2311, 22syl5eqss 3222 . . . 4  |-  ( ph  ->  dom  ( ( F  o.  ( le `  R ) )  o.  `' F )  C_  B
)
24 rncoss 4945 . . . . 5  |-  ran  (
( F  o.  ( le `  R ) )  o.  `' F ) 
C_  ran  ( F  o.  ( le `  R
) )
25 rnco2 5180 . . . . . 6  |-  ran  ( F  o.  ( le `  R ) )  =  ( F " ran  ( le `  R ) )
26 imassrn 5025 . . . . . . 7  |-  ( F
" ran  ( le `  R ) )  C_  ran  F
2726, 20syl5sseq 3226 . . . . . 6  |-  ( ph  ->  ( F " ran  ( le `  R ) )  C_  B )
2825, 27syl5eqss 3222 . . . . 5  |-  ( ph  ->  ran  ( F  o.  ( le `  R ) )  C_  B )
2924, 28syl5ss 3190 . . . 4  |-  ( ph  ->  ran  ( ( F  o.  ( le `  R ) )  o.  `' F )  C_  B
)
30 xpss12 4792 . . . 4  |-  ( ( dom  ( ( F  o.  ( le `  R ) )  o.  `' F )  C_  B  /\  ran  ( ( F  o.  ( le `  R ) )  o.  `' F )  C_  B
)  ->  ( dom  ( ( F  o.  ( le `  R ) )  o.  `' F
)  X.  ran  (
( F  o.  ( le `  R ) )  o.  `' F ) )  C_  ( B  X.  B ) )
3123, 29, 30syl2anc 642 . . 3  |-  ( ph  ->  ( dom  ( ( F  o.  ( le
`  R ) )  o.  `' F )  X.  ran  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )  C_  ( B  X.  B ) )
3210, 31syl5ss 3190 . 2  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  C_  ( B  X.  B ) )
337, 32eqsstrd 3212 1  |-  ( ph  -> 
.<_  C_  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   Rel wrel 4694   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215    "s cimas 13407
This theorem is referenced by:  xpsless  13482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-imas 13411
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