Theorem fvfullfunlem3 5863

Description: Lemma for fvfullfun 5864. Part one of the full function definition agrees with the set itself over its domain. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
fvfullfunlem3	|- (A e. dom (( _I o. F) \ ( ∼ _I o. F)) -> ((( _I o. F) \ ( ∼ _I o. F))` A) = (F` A))

Proof of Theorem fvfullfunlem3

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 5119	. . . 4 |- (Fun (F |` dom (( _I o. F) \ ( ∼ _I o. F))) <-> A.xA.yA.z((x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))y /\ x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))z) -> y = z))
2		brres 4949	. . . . . . 7 |- (x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))y <-> (xFy /\ x e. dom (( _I o. F) \ ( ∼ _I o. F))))
3		fvfullfunlem1 5861	. . . . . . . . 9 |- dom (( _I o. F) \ ( ∼ _I o. F)) = {x | E!z xFz}
4	3	abeq2i 2460	. . . . . . . 8 |- (x e. dom (( _I o. F) \ ( ∼ _I o. F)) <-> E!z xFz)
5	4	anbi2i 675	. . . . . . 7 |- ((xFy /\ x e. dom (( _I o. F) \ ( ∼ _I o. F))) <-> (xFy /\ E!z xFz))
6	2, 5	bitri 240	. . . . . 6 |- (x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))y <-> (xFy /\ E!z xFz))
7		brres 4949	. . . . . . 7 |- (x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))z <-> (xFz /\ x e. dom (( _I o. F) \ ( ∼ _I o. F))))
8		fvfullfunlem1 5861	. . . . . . . . 9 |- dom (( _I o. F) \ ( ∼ _I o. F)) = {x | E!y xFy}
9	8	abeq2i 2460	. . . . . . . 8 |- (x e. dom (( _I o. F) \ ( ∼ _I o. F)) <-> E!y xFy)
10	9	anbi2i 675	. . . . . . 7 |- ((xFz /\ x e. dom (( _I o. F) \ ( ∼ _I o. F))) <-> (xFz /\ E!y xFy))
11	7, 10	bitri 240	. . . . . 6 |- (x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))z <-> (xFz /\ E!y xFy))
12		tz6.12-1 5344	. . . . . . . . 9 |- ((xFy /\ E!y xFy) -> (F` x) = y)
13	12	adantrl 696	. . . . . . . 8 |- ((xFy /\ (xFz /\ E!y xFy)) -> (F` x) = y)
14		tz6.12-1 5344	. . . . . . . . 9 |- ((xFz /\ E!y xFy) -> (F` x) = z)
15	14	adantl 452	. . . . . . . 8 |- ((xFy /\ (xFz /\ E!y xFy)) -> (F` x) = z)
16	13, 15	eqtr3d 2387	. . . . . . 7 |- ((xFy /\ (xFz /\ E!y xFy)) -> y = z)
17	16	adantlr 695	. . . . . 6 |- (((xFy /\ E!z xFz) /\ (xFz /\ E!y xFy)) -> y = z)
18	6, 11, 17	syl2anb 465	. . . . 5 |- ((x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))y /\ x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))z) -> y = z)
19	18	gen2 1547	. . . 4 |- A.yA.z((x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))y /\ x(F |` dom (( _I o. F) \ ( ∼ _I o. F)))z) -> y = z)
20	1, 19	mpgbir 1550	. . 3 |- Fun (F |` dom (( _I o. F) \ ( ∼ _I o. F)))
21		fvfullfunlem2 5862	. . . . 5 |- (( _I o. F) \ ( ∼ _I o. F)) C_ F
22		ssdmrn 5099	. . . . . 6 |- (( _I o. F) \ ( ∼ _I o. F)) C_ (dom (( _I o. F) \ ( ∼ _I o. F)) X. ran (( _I o. F) \ ( ∼ _I o. F)))
23		ssv 3291	. . . . . . 7 |- ran (( _I o. F) \ ( ∼ _I o. F)) C_ _V
24		xpss2 4857	. . . . . . 7 |- (ran (( _I o. F) \ ( ∼ _I o. F)) C_ _V -> (dom (( _I o. F) \ ( ∼ _I o. F)) X. ran (( _I o. F) \ ( ∼ _I o. F))) C_ (dom (( _I o. F) \ ( ∼ _I o. F)) X. _V))
25	23, 24	ax-mp 5	. . . . . 6 |- (dom (( _I o. F) \ ( ∼ _I o. F)) X. ran (( _I o. F) \ ( ∼ _I o. F))) C_ (dom (( _I o. F) \ ( ∼ _I o. F)) X. _V)
26	22, 25	sstri 3281	. . . . 5 |- (( _I o. F) \ ( ∼ _I o. F)) C_ (dom (( _I o. F) \ ( ∼ _I o. F)) X. _V)
27	21, 26	ssini 3478	. . . 4 |- (( _I o. F) \ ( ∼ _I o. F)) C_ (F i^i (dom (( _I o. F) \ ( ∼ _I o. F)) X. _V))
28		df-res 4788	. . . 4 |- (F |` dom (( _I o. F) \ ( ∼ _I o. F))) = (F i^i (dom (( _I o. F) \ ( ∼ _I o. F)) X. _V))
29	27, 28	sseqtr4i 3304	. . 3 |- (( _I o. F) \ ( ∼ _I o. F)) C_ (F |` dom (( _I o. F) \ ( ∼ _I o. F)))
30		funssfv 5343	. . 3 |- ((Fun (F |` dom (( _I o. F) \ ( ∼ _I o. F))) /\ (( _I o. F) \ ( ∼ _I o. F)) C_ (F |` dom (( _I o. F) \ ( ∼ _I o. F))) /\ A e. dom (( _I o. F) \ ( ∼ _I o. F))) -> ((F |` dom (( _I o. F) \ ( ∼ _I o. F)))` A) = ((( _I o. F) \ ( ∼ _I o. F))` A))
31	20, 29, 30	mp3an12 1267	. 2 |- (A e. dom (( _I o. F) \ ( ∼ _I o. F)) -> ((F |` dom (( _I o. F) \ ( ∼ _I o. F)))` A) = ((( _I o. F) \ ( ∼ _I o. F))` A))
32		fvres 5342	. 2 |- (A e. dom (( _I o. F) \ ( ∼ _I o. F)) -> ((F |` dom (( _I o. F) \ ( ∼ _I o. F)))` A) = (F` A))
33	31, 32	eqtr3d 2387	1 |- (A e. dom (( _I o. F) \ ( ∼ _I o. F)) -> ((( _I o. F) \ ( ∼ _I o. F))` A) = (F` A))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2859   ∼ ccompl 3205   \ cdif 3206   i^i cin 3208   C_ wss 3257   class class class wbr 4639   o. ccom 4721   _I cid 4763   X. cxp 4770  dom cdm 4772  ran crn 4773   |` cres 4774  Fun wfun 4775  ` cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fv 4795

This theorem is referenced by:  fvfullfun  5864