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Theorem ssclem 14011
Description: Lemma for ssc1 14013 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 5081 . . 3  |-  dom  ( S  X.  S )  =  S
2 isssc.1 . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 fndm 5536 . . . . . . 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 5122 . . . . . 6  |-  ( H  e.  _V  ->  dom  H  e.  _V )
76adantl 453 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  e.  _V )
85, 7eqeltrrd 2510 . . . 4  |-  ( (
ph  /\  H  e.  _V )  ->  ( S  X.  S )  e. 
_V )
9 dmexg 5122 . . . 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 2520 . 2  |-  ( (
ph  /\  H  e.  _V )  ->  S  e. 
_V )
12 xpexg 4981 . . . 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 5953 . . 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 1652    e. wcel 1725   _Vcvv 2948    X. cxp 4868   dom cdm 4870    Fn wfn 5441
This theorem is referenced by:  ssc1  14013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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