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Theorem ssclem 13948
Description: Lemma for ssc1 13950 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
isssc.1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
Assertion
Ref Expression
ssclem  |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )

Proof of Theorem ssclem
StepHypRef Expression
1 dmxpid 5031 . . 3  |-  dom  ( S  X.  S )  =  S
2 isssc.1 . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 fndm 5486 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
54adantr 452 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  =  ( S  X.  S ) )
6 dmexg 5072 . . . . . 6  |-  ( H  e.  _V  ->  dom  H  e.  _V )
76adantl 453 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  e.  _V )
85, 7eqeltrrd 2464 . . . 4  |-  ( (
ph  /\  H  e.  _V )  ->  ( S  X.  S )  e. 
_V )
9 dmexg 5072 . . . 4  |-  ( ( S  X.  S )  e.  _V  ->  dom  ( S  X.  S
)  e.  _V )
108, 9syl 16 . . 3  |-  ( (
ph  /\  H  e.  _V )  ->  dom  ( S  X.  S )  e. 
_V )
111, 10syl5eqelr 2474 . 2  |-  ( (
ph  /\  H  e.  _V )  ->  S  e. 
_V )
12 xpexg 4931 . . . 4  |-  ( ( S  e.  _V  /\  S  e.  _V )  ->  ( S  X.  S
)  e.  _V )
1312anidms 627 . . 3  |-  ( S  e.  _V  ->  ( S  X.  S )  e. 
_V )
14 fnex 5902 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
152, 13, 14syl2an 464 . 2  |-  ( (
ph  /\  S  e.  _V )  ->  H  e. 
_V )
1611, 15impbida 806 1  |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    X. cxp 4818   dom cdm 4820    Fn wfn 5391
This theorem is referenced by:  ssc1  13950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404
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