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Theorem ssclem 13696
Description: Lemma for ssc1 13698 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 4898 . . 3  |-  dom  ( S  X.  S )  =  S
2 isssc.1 . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 fndm 5343 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
42, 3syl 15 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
54adantr 451 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  =  ( S  X.  S ) )
6 dmexg 4939 . . . . . 6  |-  ( H  e.  _V  ->  dom  H  e.  _V )
76adantl 452 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  e.  _V )
85, 7eqeltrrd 2358 . . . 4  |-  ( (
ph  /\  H  e.  _V )  ->  ( S  X.  S )  e. 
_V )
9 dmexg 4939 . . . 4  |-  ( ( S  X.  S )  e.  _V  ->  dom  ( S  X.  S
)  e.  _V )
108, 9syl 15 . . 3  |-  ( (
ph  /\  H  e.  _V )  ->  dom  ( S  X.  S )  e. 
_V )
111, 10syl5eqelr 2368 . 2  |-  ( (
ph  /\  H  e.  _V )  ->  S  e. 
_V )
12 xpexg 4800 . . . 4  |-  ( ( S  e.  _V  /\  S  e.  _V )  ->  ( S  X.  S
)  e.  _V )
1312anidms 626 . . 3  |-  ( S  e.  _V  ->  ( S  X.  S )  e. 
_V )
14 fnex 5741 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
152, 13, 14syl2an 463 . 2  |-  ( (
ph  /\  S  e.  _V )  ->  H  e. 
_V )
1611, 15impbida 805 1  |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    X. cxp 4687   dom cdm 4689    Fn wfn 5250
This theorem is referenced by:  ssc1  13698
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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
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