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Theorem fmptsng 6092
Description: Express a singleton function in maps-to notation. Version of fmptsn 6091 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)
Hypothesis
Ref Expression
fmptsng.1
Assertion
Ref Expression
fmptsng
Distinct variable groups:   ,   ,

Proof of Theorem fmptsng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 4043 . . . . 5
21bicomi 202 . . . 4
32anbi1i 695 . . 3
43opabbii 4516 . 2
5 elsn 4043 . . . . 5
6 eqidd 2458 . . . . . . 7
7 eqidd 2458 . . . . . . 7
8 eqeq1 2461 . . . . . . . . . 10
98adantr 465 . . . . . . . . 9
10 eqeq1 2461 . . . . . . . . . 10
11 fmptsng.1 . . . . . . . . . . 11
1211eqeq2d 2471 . . . . . . . . . 10
1310, 12sylan9bbr 700 . . . . . . . . 9
149, 13anbi12d 710 . . . . . . . 8
1514opelopabga 4765 . . . . . . 7
166, 7, 15mpbir2and 922 . . . . . 6
17 eleq1 2529 . . . . . 6
1816, 17syl5ibrcom 222 . . . . 5
195, 18syl5bi 217 . . . 4
20 elopab 4760 . . . . 5
21 opeq12 4219 . . . . . . . . . 10
2221eqeq2d 2471 . . . . . . . . 9
2311adantr 465 . . . . . . . . . . . 12
2423opeq2d 4224 . . . . . . . . . . 11
25 opex 4716 . . . . . . . . . . . 12
2625snid 4057 . . . . . . . . . . 11
2724, 26syl6eqel 2553 . . . . . . . . . 10
28 eleq1 2529 . . . . . . . . . 10
2927, 28syl5ibrcom 222 . . . . . . . . 9
3022, 29sylbid 215 . . . . . . . 8
3130impcom 430 . . . . . . 7
3231exlimivv 1723 . . . . . 6
3332a1i 11 . . . . 5
3420, 33syl5bi 217 . . . 4
3519, 34impbid 191 . . 3
3635eqrdv 2454 . 2
37 df-mpt 4512 . . 3
3837a1i 11 . 2
394, 36, 383eqtr4a 2524 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {csn 4029  <.cop 4035  {copab 4509  e.cmpt 4510
This theorem is referenced by:  mdet0pr  19094  m1detdiag  19099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-mpt 4512
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