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Theorem disjrdx 27661
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 5798 . . . . . . 7  |-  ( F : A -1-1-onto-> C  ->  F : A
--> C )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F : A --> C )
43ffvelrnda 6007 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  C )
5 f1ofveu 6265 . . . . . . 7  |-  ( ( F : A -1-1-onto-> C  /\  y  e.  C )  ->  E! x  e.  A  ( F `  x )  =  y )
61, 5sylan 469 . . . . . 6  |-  ( (
ph  /\  y  e.  C )  ->  E! x  e.  A  ( F `  x )  =  y )
7 eqcom 2463 . . . . . . 7  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
87reubii 3041 . . . . . 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 2524 . . . . 5  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  ( z  e.  D  <->  z  e.  B
) )
124, 9, 11rmoxfrd 27590 . . . 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 1718 . 2  |-  ( ph  ->  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  C  z  e.  D )
)
15 df-disj 4411 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
16 df-disj 4411 . 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 367   A.wal 1396    = wceq 1398    e. wcel 1823   E!wreu 2806   E*wrmo 2807  Disj wdisj 4410   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-disj 4411  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  volmeas  28440  carsggect  28526
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