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Theorem ssclem 15052
Description: Lemma for ssc1 15054 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 5222 . . 3  |-  dom  ( S  X.  S )  =  S
2 isssc.1 . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 fndm 5680 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
54adantr 465 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  =  ( S  X.  S ) )
6 dmexg 6716 . . . . . 6  |-  ( H  e.  _V  ->  dom  H  e.  _V )
76adantl 466 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  e.  _V )
85, 7eqeltrrd 2556 . . . 4  |-  ( (
ph  /\  H  e.  _V )  ->  ( S  X.  S )  e. 
_V )
9 dmexg 6716 . . . 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 2560 . 2  |-  ( (
ph  /\  H  e.  _V )  ->  S  e. 
_V )
12 xpexg 6587 . . . 4  |-  ( ( S  e.  _V  /\  S  e.  _V )  ->  ( S  X.  S
)  e.  _V )
1312anidms 645 . . 3  |-  ( S  e.  _V  ->  ( S  X.  S )  e. 
_V )
14 fnex 6128 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
152, 13, 14syl2an 477 . 2  |-  ( (
ph  /\  S  e.  _V )  ->  H  e. 
_V )
1611, 15impbida 830 1  |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    X. cxp 4997   dom cdm 4999    Fn wfn 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  ssc1  15054
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