Theorem actfunsnrndisj 30683

Description: The action F of extending function from B to C with new values at point I yields different functions. (Contributed by Thierry Arnoux, 9-Dec-2021.)

Hypotheses

Ref	Expression
actfunsn.1	|- ( ( ph /\ k e. C ) -> A C_ ( C ^m B ) )
actfunsn.2	|- ( ph -> C e. _V )
actfunsn.3	|- ( ph -> I e. V )
actfunsn.4	|- ( ph -> -. I e. B )
actfunsn.5	|- F = ( x e. A |-> ( x u. { <. I , k >. } ) )

Assertion

Ref	Expression
actfunsnrndisj	|- ( ph -> Disj k e. C ran F )

Distinct variable groups:   x, A   k, I, x   ph, k

Allowed substitution hints:   ph( x)   A( k)   B( x, k)   C( x, k)   F( x, k)   V( x, k)

Proof of Theorem actfunsnrndisj

Dummy variables z f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) /\ f = ( z u. { <. I , k >. } ) ) -> f = ( z u. { <. I , k >. } ) )
2	1	fveq1d 6193	. . . . . 6 |- ( ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) /\ f = ( z u. { <. I , k >. } ) ) -> ( f ` I ) = ( ( z u. { <. I , k >. } ) ` I ) )
3		actfunsn.1	. . . . . . . . . . . 12 |- ( ( ph /\ k e. C ) -> A C_ ( C ^m B ) )
4	3	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> A C_ ( C ^m B ) )
5		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> z e. A )
6	4, 5	sseldd 3604	. . . . . . . . . 10 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> z e. ( C ^m B ) )
7		elmapfn 7880	. . . . . . . . . 10 |- ( z e. ( C ^m B ) -> z Fn B )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> z Fn B )
9		actfunsn.3	. . . . . . . . . . 11 |- ( ph -> I e. V )
10	9	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> I e. V )
11		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> k e. C )
12		fnsng 5938	. . . . . . . . . 10 |- ( ( I e. V /\ k e. C ) -> { <. I , k >. } Fn { I } )
13	10, 11, 12	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> { <. I , k >. } Fn { I } )
14		actfunsn.4	. . . . . . . . . . 11 |- ( ph -> -. I e. B )
15		disjsn 4246	. . . . . . . . . . 11 |- ( ( B i^i { I } ) = (/) <-> -. I e. B )
16	14, 15	sylibr 224	. . . . . . . . . 10 |- ( ph -> ( B i^i { I } ) = (/) )
17	16	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> ( B i^i { I } ) = (/) )
18		snidg 4206	. . . . . . . . . 10 |- ( I e. V -> I e. { I } )
19	10, 18	syl 17	. . . . . . . . 9 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> I e. { I } )
20		fvun2 6270	. . . . . . . . 9 |- ( ( z Fn B /\ { <. I , k >. } Fn { I } /\ ( ( B i^i { I } ) = (/) /\ I e. { I } ) ) -> ( ( z u. { <. I , k >. } ) ` I ) = ( { <. I , k >. } ` I ) )
21	8, 13, 17, 19, 20	syl112anc 1330	. . . . . . . 8 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> ( ( z u. { <. I , k >. } ) ` I ) = ( { <. I , k >. } ` I ) )
22		fvsng 6447	. . . . . . . . 9 |- ( ( I e. V /\ k e. C ) -> ( { <. I , k >. } ` I ) = k )
23	10, 11, 22	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> ( { <. I , k >. } ` I ) = k )
24	21, 23	eqtrd 2656	. . . . . . 7 |- ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) -> ( ( z u. { <. I , k >. } ) ` I ) = k )
25	24	adantr 481	. . . . . 6 |- ( ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) /\ f = ( z u. { <. I , k >. } ) ) -> ( ( z u. { <. I , k >. } ) ` I ) = k )
26	2, 25	eqtrd 2656	. . . . 5 |- ( ( ( ( ( ph /\ k e. C ) /\ f e. ran F ) /\ z e. A ) /\ f = ( z u. { <. I , k >. } ) ) -> ( f ` I ) = k )
27		simpr 477	. . . . . 6 |- ( ( ( ph /\ k e. C ) /\ f e. ran F ) -> f e. ran F )
28		actfunsn.5	. . . . . . . 8 |- F = ( x e. A |-> ( x u. { <. I , k >. } ) )
29		uneq1 3760	. . . . . . . . 9 |- ( x = z -> ( x u. { <. I , k >. } ) = ( z u. { <. I , k >. } ) )
30	29	cbvmptv 4750	. . . . . . . 8 |- ( x e. A |-> ( x u. { <. I , k >. } ) ) = ( z e. A |-> ( z u. { <. I , k >. } ) )
31	28, 30	eqtri 2644	. . . . . . 7 |- F = ( z e. A |-> ( z u. { <. I , k >. } ) )
32		vex 3203	. . . . . . . 8 |- z e. _V
33		snex 4908	. . . . . . . 8 |- { <. I , k >. } e. _V
34	32, 33	unex 6956	. . . . . . 7 |- ( z u. { <. I , k >. } ) e. _V
35	31, 34	elrnmpti 5376	. . . . . 6 |- ( f e. ran F <-> E. z e. A f = ( z u. { <. I , k >. } ) )
36	27, 35	sylib 208	. . . . 5 |- ( ( ( ph /\ k e. C ) /\ f e. ran F ) -> E. z e. A f = ( z u. { <. I , k >. } ) )
37	26, 36	r19.29a 3078	. . . 4 |- ( ( ( ph /\ k e. C ) /\ f e. ran F ) -> ( f ` I ) = k )
38	37	ralrimiva 2966	. . 3 |- ( ( ph /\ k e. C ) -> A. f e. ran F ( f ` I ) = k )
39	38	ralrimiva 2966	. 2 |- ( ph -> A. k e. C A. f e. ran F ( f ` I ) = k )
40		invdisj 4638	. 2 |- ( A. k e. C A. f e. ran F ( f ` I ) = k -> Disj k e. C ran F )
41	39, 40	syl 17	1 |- ( ph -> Disj k e. C ran F )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183  Disj wdisj 4620   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-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-rmo 2920  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-iun 4522  df-disj 4621  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  breprexplema  30708