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Theorem off2 25981
Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
Hypotheses
Ref Expression
off2.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
off2.2  |-  ( ph  ->  F : A --> S )
off2.3  |-  ( ph  ->  G : B --> T )
off2.4  |-  ( ph  ->  A  e.  V )
off2.5  |-  ( ph  ->  B  e.  W )
off2.6  |-  ( ph  ->  ( A  i^i  B
)  =  C )
Assertion
Ref Expression
off2  |-  ( ph  ->  ( F  oF R G ) : C --> U )
Distinct variable groups:    y, G    x, y, ph    x, S, y   
x, T, y    x, F, y    x, R, y   
x, U, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    G( x)    V( x, y)    W( x, y)

Proof of Theorem off2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 off2.2 . . . . . 6  |-  ( ph  ->  F : A --> S )
21adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  C )  ->  F : A --> S )
3 off2.6 . . . . . . 7  |-  ( ph  ->  ( A  i^i  B
)  =  C )
4 inss1 3591 . . . . . . 7  |-  ( A  i^i  B )  C_  A
53, 4syl6eqssr 3428 . . . . . 6  |-  ( ph  ->  C  C_  A )
65sselda 3377 . . . . 5  |-  ( (
ph  /\  z  e.  C )  ->  z  e.  A )
72, 6ffvelrnd 5865 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off2.3 . . . . . 6  |-  ( ph  ->  G : B --> T )
98adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  C )  ->  G : B --> T )
10 inss2 3592 . . . . . . 7  |-  ( A  i^i  B )  C_  B
113, 10syl6eqssr 3428 . . . . . 6  |-  ( ph  ->  C  C_  B )
1211sselda 3377 . . . . 5  |-  ( (
ph  /\  z  e.  C )  ->  z  e.  B )
139, 12ffvelrnd 5865 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( G `  z )  e.  T )
14 off2.1 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
1514ralrimivva 2829 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 465 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 proplem2 14648 . . . 4  |-  ( ( ( ( F `  z )  e.  S  /\  ( G `  z
)  e.  T )  /\  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U
)  ->  ( ( F `  z ) R ( G `  z ) )  e.  U )
187, 13, 16, 17syl21anc 1217 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
19 eqid 2443 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2018, 19fmptd 5888 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
21 ffn 5580 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
221, 21syl 16 . . . . 5  |-  ( ph  ->  F  Fn  A )
23 ffn 5580 . . . . . 6  |-  ( G : B --> T  ->  G  Fn  B )
248, 23syl 16 . . . . 5  |-  ( ph  ->  G  Fn  B )
25 off2.4 . . . . 5  |-  ( ph  ->  A  e.  V )
26 off2.5 . . . . 5  |-  ( ph  ->  B  e.  W )
27 eqid 2443 . . . . 5  |-  ( A  i^i  B )  =  ( A  i^i  B
)
28 eqidd 2444 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
29 eqidd 2444 . . . . 5  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3022, 24, 25, 26, 27, 28, 29offval 6348 . . . 4  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  ( A  i^i  B ) 
|->  ( ( F `  z ) R ( G `  z ) ) ) )
313mpteq1d 4394 . . . 4  |-  ( ph  ->  ( z  e.  ( A  i^i  B ) 
|->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `  z
) R ( G `
 z ) ) ) )
3230, 31eqtrd 2475 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3332feq1d 5567 . 2  |-  ( ph  ->  ( ( F  oF R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3420, 33mpbird 232 1  |-  ( ph  ->  ( F  oF R G ) : C --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736    i^i cin 3348    e. cmpt 4371    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    oFcof 6339
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-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341
This theorem is referenced by: (None)
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