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Theorem ialgrlem1st 11298
Description: Lemma for ialgr0 11300. Expressing algrflemg 5995 in a form suitable for theorems such as iseq1 9871 or iseqfcl 9874. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypothesis
Ref Expression
ialgrlem1st.f  |-  ( ph  ->  F : S --> S )
Assertion
Ref Expression
ialgrlem1st  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )

Proof of Theorem ialgrlem1st
StepHypRef Expression
1 algrflemg 5995 . . 3  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( F  o.  1st ) y )  =  ( F `
 x ) )
21adantl 271 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  =  ( F `
 x ) )
3 ialgrlem1st.f . . . 4  |-  ( ph  ->  F : S --> S )
43adantr 270 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  F : S --> S )
5 simprl 498 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
64, 5ffvelrnd 5435 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( F `  x
)  e.  S )
72, 6eqeltrd 2164 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    o. ccom 4442   -->wf 5011   ` cfv 5015  (class class class)co 5652   1stc1st 5909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-ov 5655  df-1st 5911
This theorem is referenced by:  ialgr0  11300  ialgrf  11301  ialgrp1  11302
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