Theorem fndmeng 6776

Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
fndmeng	|- ( ( F Fn A /\ A e. C ) -> A ~~ F )

Proof of Theorem fndmeng

Step	Hyp	Ref	Expression
1		fnex 5707	. . 3 |- ( ( F Fn A /\ A e. C ) -> F e. _V )
2		fnfun 5285	. . . 4 |- ( F Fn A -> Fun F )
3	2	adantr 274	. . 3 |- ( ( F Fn A /\ A e. C ) -> Fun F )
4		fundmeng 6773	. . 3 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
5	1, 3, 4	syl2anc 409	. 2 |- ( ( F Fn A /\ A e. C ) -> dom F ~~ F )
6		fndm 5287	. . . 4 |- ( F Fn A -> dom F = A )
7	6	breq1d 3992	. . 3 |- ( F Fn A -> ( dom F ~~ F <-> A ~~ F ) )
8	7	adantr 274	. 2 |- ( ( F Fn A /\ A e. C ) -> ( dom F ~~ F <-> A ~~ F ) )
9	5, 8	mpbid 146	1 |- ( ( F Fn A /\ A e. C ) -> A ~~ F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   dom cdm 4604   Fun wfun 5182   Fn wfn 5183   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-en 6707

This theorem is referenced by:  fihashfn  10713