Theorem disjpreima 29397

Description: A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)

Assertion

Ref	Expression
disjpreima	|- ( ( Fun F /\ Disj x e. A B ) -> Disj x e. A ( `' F " B ) )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   B( x)

Proof of Theorem disjpreima

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

Step	Hyp	Ref	Expression
1		inpreima 6342	. . . . . . . . 9 |- ( Fun F -> ( `' F " ( [_ y / x ]_ B i^i [_ z / x ]_ B ) ) = ( ( `' F " [_ y / x ]_ B ) i^i ( `' F " [_ z / x ]_ B ) ) )
2		imaeq2 5462	. . . . . . . . . 10 |- ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) -> ( `' F " ( [_ y / x ]_ B i^i [_ z / x ]_ B ) ) = ( `' F " (/) ) )
3		ima0 5481	. . . . . . . . . 10 |- ( `' F " (/) ) = (/)
4	2, 3	syl6eq 2672	. . . . . . . . 9 |- ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) -> ( `' F " ( [_ y / x ]_ B i^i [_ z / x ]_ B ) ) = (/) )
5	1, 4	sylan9req 2677	. . . . . . . 8 |- ( ( Fun F /\ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> ( ( `' F " [_ y / x ]_ B ) i^i ( `' F " [_ z / x ]_ B ) ) = (/) )
6	5	ex 450	. . . . . . 7 |- ( Fun F -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) -> ( ( `' F " [_ y / x ]_ B ) i^i ( `' F " [_ z / x ]_ B ) ) = (/) ) )
7		csbima12 5483	. . . . . . . . . 10 |- [_ y / x ]_ ( `' F " B ) = ( [_ y / x ]_ `' F " [_ y / x ]_ B )
8		vex 3203	. . . . . . . . . . . 12 |- y e. _V
9		csbconstg 3546	. . . . . . . . . . . 12 |- ( y e. _V -> [_ y / x ]_ `' F = `' F )
10	8, 9	ax-mp 5	. . . . . . . . . . 11 |- [_ y / x ]_ `' F = `' F
11	10	imaeq1i 5463	. . . . . . . . . 10 |- ( [_ y / x ]_ `' F " [_ y / x ]_ B ) = ( `' F " [_ y / x ]_ B )
12	7, 11	eqtri 2644	. . . . . . . . 9 |- [_ y / x ]_ ( `' F " B ) = ( `' F " [_ y / x ]_ B )
13		csbima12 5483	. . . . . . . . . 10 |- [_ z / x ]_ ( `' F " B ) = ( [_ z / x ]_ `' F " [_ z / x ]_ B )
14		vex 3203	. . . . . . . . . . . 12 |- z e. _V
15		csbconstg 3546	. . . . . . . . . . . 12 |- ( z e. _V -> [_ z / x ]_ `' F = `' F )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- [_ z / x ]_ `' F = `' F
17	16	imaeq1i 5463	. . . . . . . . . 10 |- ( [_ z / x ]_ `' F " [_ z / x ]_ B ) = ( `' F " [_ z / x ]_ B )
18	13, 17	eqtri 2644	. . . . . . . . 9 |- [_ z / x ]_ ( `' F " B ) = ( `' F " [_ z / x ]_ B )
19	12, 18	ineq12i 3812	. . . . . . . 8 |- ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = ( ( `' F " [_ y / x ]_ B ) i^i ( `' F " [_ z / x ]_ B ) )
20	19	eqeq1i 2627	. . . . . . 7 |- ( ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) <-> ( ( `' F " [_ y / x ]_ B ) i^i ( `' F " [_ z / x ]_ B ) ) = (/) )
21	6, 20	syl6ibr 242	. . . . . 6 |- ( Fun F -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) -> ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) ) )
22	21	orim2d 885	. . . . 5 |- ( Fun F -> ( ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> ( y = z \/ ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) ) ) )
23	22	ralimdv 2963	. . . 4 |- ( Fun F -> ( A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> A. z e. A ( y = z \/ ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) ) ) )
24	23	ralimdv 2963	. . 3 |- ( Fun F -> ( A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) ) ) )
25		disjors 4635	. . 3 |- (Disj x e. A B <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
26		disjors 4635	. . 3 |- (Disj x e. A ( `' F " B ) <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ ( `' F " B ) i^i [_ z / x ]_ ( `' F " B ) ) = (/) ) )
27	24, 25, 26	3imtr4g 285	. 2 |- ( Fun F -> (Disj x e. A B -> Disj x e. A ( `' F " B ) ) )
28	27	imp 445	1 |- ( ( Fun F /\ Disj x e. A B ) -> Disj x e. A ( `' F " B ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   `'ccnv 5113   "cima 5117   Fun wfun 5882

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-ral 2917  df-rex 2918  df-reu 2919  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-sn 4178  df-pr 4180  df-op 4184  df-disj 4621  df-br 4654  df-opab 4713  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-fun 5890

This theorem is referenced by:  sibfof  30402  dstrvprob  30533