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Theorem fmptsng 6091
 Description: Express a singleton function in maps-to notation. Version of fmptsn 6090 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:   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem fmptsng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 4007 . . . . 5
21bicomi 205 . . . 4
32anbi1i 699 . . 3
43opabbii 4481 . 2
5 elsn 4007 . . . . 5
6 eqidd 2421 . . . . . . 7
7 eqidd 2421 . . . . . . 7
8 eqeq1 2424 . . . . . . . . . 10
98adantr 466 . . . . . . . . 9
10 eqeq1 2424 . . . . . . . . . 10
11 fmptsng.1 . . . . . . . . . . 11
1211eqeq2d 2434 . . . . . . . . . 10
1310, 12sylan9bbr 705 . . . . . . . . 9
149, 13anbi12d 715 . . . . . . . 8
1514opelopabga 4725 . . . . . . 7
166, 7, 15mpbir2and 930 . . . . . 6
17 eleq1 2492 . . . . . 6
1816, 17syl5ibrcom 225 . . . . 5
195, 18syl5bi 220 . . . 4
20 elopab 4720 . . . . 5
21 opeq12 4183 . . . . . . . . . 10
2221eqeq2d 2434 . . . . . . . . 9
2311adantr 466 . . . . . . . . . . . 12
2423opeq2d 4188 . . . . . . . . . . 11
25 opex 4677 . . . . . . . . . . . 12
2625snid 4021 . . . . . . . . . . 11
2724, 26syl6eqel 2516 . . . . . . . . . 10
28 eleq1 2492 . . . . . . . . . 10
2927, 28syl5ibrcom 225 . . . . . . . . 9
3022, 29sylbid 218 . . . . . . . 8
3130impcom 431 . . . . . . 7
3231exlimivv 1767 . . . . . 6
3332a1i 11 . . . . 5
3420, 33syl5bi 220 . . . 4
3519, 34impbid 193 . . 3
3635eqrdv 2417 . 2
37 df-mpt 4477 . . 3
3837a1i 11 . 2
394, 36, 383eqtr4a 2487 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437  wex 1659   wcel 1867  csn 3993  cop 3999  copab 4474   cmpt 4475 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-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 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-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-opab 4476  df-mpt 4477 This theorem is referenced by:  mdet0pr  19554  m1detdiag  19559
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