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Theorem funcnv5mpt 25988
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
Hypotheses
Ref Expression
funcnvmpt.0  |-  F/ x ph
funcnvmpt.1  |-  F/_ x A
funcnvmpt.2  |-  F/_ x F
funcnvmpt.3  |-  F  =  ( x  e.  A  |->  B )
funcnvmpt.4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
funcnv5mpt.1  |-  ( x  =  z  ->  B  =  C )
Assertion
Ref Expression
funcnv5mpt  |-  ( ph  ->  ( Fun  `' F  <->  A. x  e.  A  A. z  e.  A  (
x  =  z  \/  B  =/=  C ) ) )
Distinct variable groups:    x, z    ph, z    z, A    z, B    x, C
Allowed substitution hints:    ph( x)    A( x)    B( x)    C( z)    F( x, z)    V( x, z)

Proof of Theorem funcnv5mpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funcnvmpt.0 . . 3  |-  F/ x ph
2 funcnvmpt.1 . . 3  |-  F/_ x A
3 funcnvmpt.2 . . 3  |-  F/_ x F
4 funcnvmpt.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 funcnvmpt.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
61, 2, 3, 4, 5funcnvmpt 25987 . 2  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
7 nne 2612 . . . . . . . . 9  |-  ( -.  B  =/=  C  <->  B  =  C )
8 eqvincg 25859 . . . . . . . . . 10  |-  ( B  e.  V  ->  ( B  =  C  <->  E. y
( y  =  B  /\  y  =  C ) ) )
95, 8syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  C  <->  E. y
( y  =  B  /\  y  =  C ) ) )
107, 9syl5bb 257 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  B  =/=  C  <->  E. y ( y  =  B  /\  y  =  C ) ) )
1110imbi1d 317 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( -.  B  =/= 
C  ->  x  =  z )  <->  ( E. y ( y  =  B  /\  y  =  C )  ->  x  =  z ) ) )
12 orcom 387 . . . . . . . 8  |-  ( ( x  =  z  \/  B  =/=  C )  <-> 
( B  =/=  C  \/  x  =  z
) )
13 df-or 370 . . . . . . . 8  |-  ( ( B  =/=  C  \/  x  =  z )  <->  ( -.  B  =/=  C  ->  x  =  z ) )
1412, 13bitri 249 . . . . . . 7  |-  ( ( x  =  z  \/  B  =/=  C )  <-> 
( -.  B  =/= 
C  ->  x  =  z ) )
15 19.23v 1910 . . . . . . 7  |-  ( A. y ( ( y  =  B  /\  y  =  C )  ->  x  =  z )  <->  ( E. y ( y  =  B  /\  y  =  C )  ->  x  =  z ) )
1611, 14, 153bitr4g 288 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  =  z  \/  B  =/=  C
)  <->  A. y ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) ) )
1716ralbidv 2735 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( A. z  e.  A  ( x  =  z  \/  B  =/=  C
)  <->  A. z  e.  A  A. y ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) ) )
18 ralcom4 2991 . . . . 5  |-  ( A. z  e.  A  A. y ( ( y  =  B  /\  y  =  C )  ->  x  =  z )  <->  A. y A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) )
1917, 18syl6bb 261 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( A. z  e.  A  ( x  =  z  \/  B  =/=  C
)  <->  A. y A. z  e.  A  ( (
y  =  B  /\  y  =  C )  ->  x  =  z ) ) )
201, 19ralbida 2729 . . 3  |-  ( ph  ->  ( A. x  e.  A  A. z  e.  A  ( x  =  z  \/  B  =/= 
C )  <->  A. x  e.  A  A. y A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) ) )
21 nfcv 2579 . . . . . 6  |-  F/_ z A
22 nfv 1673 . . . . . 6  |-  F/ x  y  =  C
23 funcnv5mpt.1 . . . . . . 7  |-  ( x  =  z  ->  B  =  C )
2423eqeq2d 2454 . . . . . 6  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  C ) )
252, 21, 22, 24rmo4f 25881 . . . . 5  |-  ( E* x  e.  A  y  =  B  <->  A. x  e.  A  A. z  e.  A  ( (
y  =  B  /\  y  =  C )  ->  x  =  z ) )
2625albii 1610 . . . 4  |-  ( A. y E* x  e.  A  y  =  B  <->  A. y A. x  e.  A  A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) )
27 ralcom4 2991 . . . 4  |-  ( A. x  e.  A  A. y A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z )  <->  A. y A. x  e.  A  A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) )
2826, 27bitr4i 252 . . 3  |-  ( A. y E* x  e.  A  y  =  B  <->  A. x  e.  A  A. y A. z  e.  A  ( ( y  =  B  /\  y  =  C )  ->  x  =  z ) )
2920, 28syl6bbr 263 . 2  |-  ( ph  ->  ( A. x  e.  A  A. z  e.  A  ( x  =  z  \/  B  =/= 
C )  <->  A. y E* x  e.  A  y  =  B )
)
306, 29bitr4d 256 1  |-  ( ph  ->  ( Fun  `' F  <->  A. x  e.  A  A. z  e.  A  (
x  =  z  \/  B  =/=  C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586   F/wnf 1589    e. wcel 1756   F/_wnfc 2566    =/= wne 2606   A.wral 2715   E*wrmo 2718    e. cmpt 4350   `'ccnv 4839   Fun wfun 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  funcnv4mpt  25989
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