Theorem fndmeng 7092

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 5931	. . 3 |- ( ( F Fn A /\ A e. C ) -> F e. _V )
2		fnfun 5476	. . . 4 |- ( F Fn A -> Fun F )
3	2	adantr 276	. . 3 |- ( ( F Fn A /\ A e. C ) -> Fun F )
4		fundmeng 7089	. . 3 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
5	1, 3, 4	syl2anc 415	. 2 |- ( ( F Fn A /\ A e. C ) -> dom F ~~ F )
6		fndm 5478	. . . 4 |- ( F Fn A -> dom F = A )
7	6	breq1d 4138	. . 3 |- ( F Fn A -> ( dom F ~~ F <-> A ~~ F ) )
8	7	adantr 276	. 2 |- ( ( F Fn A /\ A e. C ) -> ( dom F ~~ F <-> A ~~ F ) )
9	5, 8	mpbid 147	1 |- ( ( F Fn A /\ A e. C ) -> A ~~ F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   _Vcvv 2821   class class class wbr 4128   dom cdm 4772   Fun wfun 5369   Fn wfn 5370   ~~ cen 7014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-en 7017

This theorem is referenced by:  fihashfn  11223