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Theorem mpt2fvex 5771
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpt2.1  F  ,  |->  C
Assertion
Ref Expression
mpt2fvex  C  V  R  W  S  X  R F S 
_V
Distinct variable groups:   ,,   ,,
Allowed substitution hints:    C(,)    R(,)    S(,)    F(,)    V(,)    W(,)    X(,)

Proof of Theorem mpt2fvex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ov 5458 . 2  R F S  F `  <. R ,  S >.
2 elex 2560 . . . . . . . . 9  C  V  C  _V
32alimi 1341 . . . . . . . 8  C  V  C 
_V
4 vex 2554 . . . . . . . . 9 
_V
5 2ndexg 5737 . . . . . . . . 9  _V  2nd ` 
_V
6 nfcv 2175 . . . . . . . . . 10  F/_ 2nd `
7 nfcsb1v 2876 . . . . . . . . . . 11  F/_ [_ 2nd `  ]_ C
87nfel1 2185 . . . . . . . . . 10  F/
[_ 2nd `  ]_ C  _V
9 csbeq1a 2854 . . . . . . . . . . 11  2nd `  C  [_ 2nd `  ]_ C
109eleq1d 2103 . . . . . . . . . 10  2nd `  C 
_V 
[_ 2nd `  ]_ C  _V
116, 8, 10spcgf 2629 . . . . . . . . 9  2nd `  _V  C  _V  [_ 2nd `  ]_ C  _V
124, 5, 11mp2b 8 . . . . . . . 8  C  _V  [_ 2nd `  ]_ C  _V
133, 12syl 14 . . . . . . 7  C  V  [_ 2nd `  ]_ C  _V
1413alimi 1341 . . . . . 6  C  V  [_ 2nd `  ]_ C  _V
15 1stexg 5736 . . . . . . 7  _V  1st ` 
_V
16 nfcv 2175 . . . . . . . 8  F/_ 1st `
17 nfcsb1v 2876 . . . . . . . . 9  F/_ [_ 1st `  ]_ [_ 2nd `  ]_ C
1817nfel1 2185 . . . . . . . 8  F/ [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
19 csbeq1a 2854 . . . . . . . . 9  1st `  [_ 2nd `  ]_ C  [_ 1st `  ]_ [_ 2nd `  ]_ C
2019eleq1d 2103 . . . . . . . 8  1st `  [_ 2nd `  ]_ C  _V  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
2116, 18, 20spcgf 2629 . . . . . . 7  1st `  _V  [_ 2nd `  ]_ C  _V  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
224, 15, 21mp2b 8 . . . . . 6  [_ 2nd `  ]_ C  _V  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
2314, 22syl 14 . . . . 5  C  V  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
2423alrimiv 1751 . . . 4  C  V  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
25243ad2ant1 924 . . 3  C  V  R  W  S  X  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
26 opexg 3955 . . . 4  R  W  S  X  <. R ,  S >.  _V
27263adant1 921 . . 3  C  V  R  W  S  X  <. R ,  S >.  _V
28 fmpt2.1 . . . . 5  F  ,  |->  C
29 mpt2mptsx 5765 . . . . 5  ,  |->  C 
U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
3028, 29eqtri 2057 . . . 4  F  U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
3130mptfvex 5199 . . 3  [_ 1st `  ]_
[_ 2nd `  ]_ C  _V  <. R ,  S >.  _V  F `  <. R ,  S >.  _V
3225, 27, 31syl2anc 391 . 2  C  V  R  W  S  X  F `  <. R ,  S >.  _V
331, 32syl5eqel 2121 1  C  V  R  W  S  X  R F S 
_V
Colors of variables: wff set class
Syntax hints:   wi 4   w3a 884  wal 1240   wceq 1242   wcel 1390   _Vcvv 2551   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648    |-> cmpt 3809    X. cxp 4286   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2fvexi  5774  oaexg  5967  omexg  5970  oeiexg  5972
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