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Theorem nfimad 5204
Description: Deduction version of bound-variable hypothesis builder nfima 5203. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfimad.2  |-  ( ph  -> 
F/_ x A )
nfimad.3  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfimad  |-  ( ph  -> 
F/_ x ( A
" B ) )

Proof of Theorem nfimad
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2576 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2576 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfima 5203 . 2  |-  F/_ x
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)
4 nfimad.2 . . 3  |-  ( ph  -> 
F/_ x A )
5 nfimad.3 . . 3  |-  ( ph  -> 
F/_ x B )
6 nfnfc1 2574 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2574 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1846 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 3095 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109imaeq1d 5194 . . . . 5  |-  ( F/_ x A  ->  ( { z  |  A. x  z  e.  A } " { z  |  A. x  z  e.  B } )  =  ( A " { z  |  A. x  z  e.  B } ) )
11 abidnf 3095 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211imaeq2d 5195 . . . . 5  |-  ( F/_ x B  ->  ( A
" { z  | 
A. x  z  e.  B } )  =  ( A " B
) )
1310, 12sylan9eq 2487 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( { z  | 
A. x  z  e.  A } " {
z  |  A. x  z  e.  B }
)  =  ( A
" B ) )
148, 13nfceqdf 2570 . . 3  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( F/_ x ( { z  |  A. x  z  e.  A } " { z  |  A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
154, 5, 14syl2anc 643 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  A } " { z  | 
A. x  z  e.  B } )  <->  F/_ x ( A " B ) ) )
163, 15mpbii 203 1  |-  ( ph  -> 
F/_ x ( A
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    e. wcel 1725   {cab 2421   F/_wnfc 2558   "cima 4873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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