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Theorem disjrdx 26077
Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Hypotheses
Ref Expression
disjrdx.1  |-  ( ph  ->  F : A -1-1-onto-> C )
disjrdx.2  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  D  =  B )
Assertion
Ref Expression
disjrdx  |-  ( ph  ->  (Disj  x  e.  A  B 
<-> Disj  y  e.  C  D
) )
Distinct variable groups:    x, y, A    y, B    x, C, y    x, D    x, F, y    ph, x, y
Allowed substitution hints:    B( x)    D( y)

Proof of Theorem disjrdx
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 disjrdx.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> C )
2 f1of 5742 . . . . . . 7  |-  ( F : A -1-1-onto-> C  ->  F : A
--> C )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F : A --> C )
43ffvelrnda 5945 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  C )
5 f1ofveu 6188 . . . . . . 7  |-  ( ( F : A -1-1-onto-> C  /\  y  e.  C )  ->  E! x  e.  A  ( F `  x )  =  y )
61, 5sylan 471 . . . . . 6  |-  ( (
ph  /\  y  e.  C )  ->  E! x  e.  A  ( F `  x )  =  y )
7 eqcom 2460 . . . . . . 7  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
87reubii 3006 . . . . . 6  |-  ( E! x  e.  A  ( F `  x )  =  y  <->  E! x  e.  A  y  =  ( F `  x ) )
96, 8sylib 196 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  E! x  e.  A  y  =  ( F `  x ) )
10 disjrdx.2 . . . . . 6  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  D  =  B )
1110eleq2d 2521 . . . . 5  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  ( z  e.  D  <->  z  e.  B
) )
124, 9, 11rmoxfrd 26022 . . . 4  |-  ( ph  ->  ( E* y  e.  C  z  e.  D  <->  E* x  e.  A  z  e.  B ) )
1312bicomd 201 . . 3  |-  ( ph  ->  ( E* x  e.  A  z  e.  B  <->  E* y  e.  C  z  e.  D ) )
1413albidv 1680 . 2  |-  ( ph  ->  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  C  z  e.  D )
)
15 df-disj 4364 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
16 df-disj 4364 . 2  |-  (Disj  y  e.  C  D  <->  A. z E* y  e.  C  z  e.  D )
1714, 15, 163bitr4g 288 1  |-  ( ph  ->  (Disj  x  e.  A  B 
<-> Disj  y  e.  C  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   E!wreu 2797   E*wrmo 2798  Disj wdisj 4363   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-disj 4364  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527
This theorem is referenced by:  volmeas  26784
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