Theorem fndmeng 8034

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 6481	. . 3 |- ( ( F Fn A /\ A e. C ) -> F e. _V )
2		fnfun 5988	. . . 4 |- ( F Fn A -> Fun F )
3	2	adantr 481	. . 3 |- ( ( F Fn A /\ A e. C ) -> Fun F )
4		fundmeng 8031	. . 3 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
5	1, 3, 4	syl2anc 693	. 2 |- ( ( F Fn A /\ A e. C ) -> dom F ~~ F )
6		fndm 5990	. . . 4 |- ( F Fn A -> dom F = A )
7	6	breq1d 4663	. . 3 |- ( F Fn A -> ( dom F ~~ F <-> A ~~ F ) )
8	7	adantr 481	. 2 |- ( ( F Fn A /\ A e. C ) -> ( dom F ~~ F <-> A ~~ F ) )
9	5, 8	mpbid 222	1 |- ( ( F Fn A /\ A e. C ) -> A ~~ F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956

This theorem is referenced by:  tskcard  9603  hashfn  13164