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Theorem funcnv5mpt 27487
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 27486 . 2  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
7 nne 2644 . . . . . . . . 9  |-  ( -.  B  =/=  C  <->  B  =  C )
8 eqvincg 27350 . . . . . . . . . 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 1747 . . . . . . 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 2882 . . . . 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 3114 . . . . 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 2876 . . 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 2605 . . . . . 6  |-  F/_ z A
22 nfv 1694 . . . . . 6  |-  F/ x  y  =  C
23 funcnv5mpt.1 . . . . . . 7  |-  ( x  =  z  ->  B  =  C )
2423eqeq2d 2457 . . . . . 6  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  C ) )
252, 21, 22, 24rmo4f 27372 . . . . 5  |-  ( E* x  e.  A  y  =  B  <->  A. x  e.  A  A. z  e.  A  ( (
y  =  B  /\  y  =  C )  ->  x  =  z ) )
2625albii 1627 . . . 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 3114 . . . 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 1381    = wceq 1383   E.wex 1599   F/wnf 1603    e. wcel 1804   F/_wnfc 2591    =/= wne 2638   A.wral 2793   E*wrmo 2796    |-> cmpt 4495   `'ccnv 4988   Fun wfun 5572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by:  funcnv4mpt  27488
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