Theorem fndmeng 6824

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 5751	. . 3 |- ( ( F Fn A /\ A e. C ) -> F e. _V )
2		fnfun 5325	. . . 4 |- ( F Fn A -> Fun F )
3	2	adantr 276	. . 3 |- ( ( F Fn A /\ A e. C ) -> Fun F )
4		fundmeng 6821	. . 3 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
5	1, 3, 4	syl2anc 411	. 2 |- ( ( F Fn A /\ A e. C ) -> dom F ~~ F )
6		fndm 5327	. . . 4 |- ( F Fn A -> dom F = A )
7	6	breq1d 4025	. . 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 2158   _Vcvv 2749   class class class wbr 4015   dom cdm 4638   Fun wfun 5222   Fn wfn 5223   ~~ cen 6752

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-en 6755

This theorem is referenced by:  fihashfn  10794