Theorem fndmeng 6788

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 5718	. . 3 |- ( ( F Fn A /\ A e. C ) -> F e. _V )
2		fnfun 5295	. . . 4 |- ( F Fn A -> Fun F )
3	2	adantr 274	. . 3 |- ( ( F Fn A /\ A e. C ) -> Fun F )
4		fundmeng 6785	. . 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 5297	. . . 4 |- ( F Fn A -> dom F = A )
7	6	breq1d 3999	. . 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 2141   _Vcvv 2730   class class class wbr 3989   dom cdm 4611   Fun wfun 5192   Fn wfn 5193   ~~ cen 6716

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-en 6719

This theorem is referenced by:  fihashfn  10735