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Theorem ssclem 15668
Description: Lemma for ssc1 15670 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 5065 . . 3  |-  dom  ( S  X.  S )  =  S
2 isssc.1 . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 fndm 5684 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
42, 3syl 17 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
54adantr 466 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  =  ( S  X.  S ) )
6 dmexg 6729 . . . . . 6  |-  ( H  e.  _V  ->  dom  H  e.  _V )
76adantl 467 . . . . 5  |-  ( (
ph  /\  H  e.  _V )  ->  dom  H  e.  _V )
85, 7eqeltrrd 2509 . . . 4  |-  ( (
ph  /\  H  e.  _V )  ->  ( S  X.  S )  e. 
_V )
9 dmexg 6729 . . . 4  |-  ( ( S  X.  S )  e.  _V  ->  dom  ( S  X.  S
)  e.  _V )
108, 9syl 17 . . 3  |-  ( (
ph  /\  H  e.  _V )  ->  dom  ( S  X.  S )  e. 
_V )
111, 10syl5eqelr 2513 . 2  |-  ( (
ph  /\  H  e.  _V )  ->  S  e. 
_V )
12 sqxpexg 6601 . . 3  |-  ( S  e.  _V  ->  ( S  X.  S )  e. 
_V )
13 fnex 6138 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
142, 12, 13syl2an 479 . 2  |-  ( (
ph  /\  S  e.  _V )  ->  H  e. 
_V )
1511, 14impbida 840 1  |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    X. cxp 4843   dom cdm 4845    Fn wfn 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600
This theorem is referenced by:  ssc1  15670
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