Theorem fndmeng 13598

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 4535	. . 3 |- ((F Fn A /\ A e. C) -> F e. _V)
2		fnfun 4510	. . . 4 |- (F Fn A -> Fun F)
3	2	adantr 425	. . 3 |- ((F Fn A /\ A e. C) -> Fun F)
4		funeq 4441	. . . . . 6 |- (F = if(F e. _V, F, (/)) -> (Fun F <-> Fun if(F e. _V, F, (/))))
5		dmeq 4157	. . . . . . 7 |- (F = if(F e. _V, F, (/)) -> dom F = dom if(F e. _V, F, (/)))
6		id 73	. . . . . . 7 |- (F = if(F e. _V, F, (/)) -> F = if(F e. _V, F, (/)))
7	5, 6	breq12d 3351	. . . . . 6 |- (F = if(F e. _V, F, (/)) -> (dom F ~~ F <-> dom if(F e. _V, F, (/)) ~~ if(F e. _V, F, (/))))
8	4, 7	imbi12d 688	. . . . 5 |- (F = if(F e. _V, F, (/)) -> ((Fun F -> dom F ~~ F) <-> (Fun if(F e. _V, F, (/)) -> dom if(F e. _V, F, (/)) ~~ if(F e. _V, F, (/)))))
9		0ex 3446	. . . . . . 7 |- (/) e. _V
10	9	elimel 3025	. . . . . 6 |- if(F e. _V, F, (/)) e. _V
11	10	fundmen 5487	. . . . 5 |- (Fun if(F e. _V, F, (/)) -> dom if(F e. _V, F, (/)) ~~ if(F e. _V, F, (/)))
12	8, 11	dedth 3011	. . . 4 |- (F e. _V -> (Fun F -> dom F ~~ F))
13	12	imp 377	. . 3 |- ((F e. _V /\ Fun F) -> dom F ~~ F)
14	1, 3, 13	syl11anc 524	. 2 |- ((F Fn A /\ A e. C) -> dom F ~~ F)
15		fndm 4512	. . . 4 |- (F Fn A -> dom F = A)
16	15	breq1d 3348	. . 3 |- (F Fn A -> (dom F ~~ F <-> A ~~ F))
17	16	adantr 425	. 2 |- ((F Fn A /\ A e. C) -> (dom F ~~ F <-> A ~~ F))
18	14, 17	mpbid 212	1 |- ((F Fn A /\ A e. C) -> A ~~ F)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982   class class class wbr 3338  dom cdm 3986  Fun wfun 3992   Fn wfn 3993   ~~ cen 5423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427

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