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Theorem ialgrlem1st 11734
Description: Lemma for ialgr0 11736. Expressing algrflemg 6127 in a form suitable for theorems such as seq3-1 10245 or seqf 10246. (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 6127 . . 3  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( F  o.  1st ) y )  =  ( F `
 x ) )
21adantl 275 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  =  ( F `
 x ) )
3 ialgrlem1st.f . . . 4  |-  ( ph  ->  F : S --> S )
43adantr 274 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  F : S --> S )
5 simprl 520 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
64, 5ffvelrnd 5556 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( F `  x
)  e.  S )
72, 6eqeltrd 2216 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 103    = wceq 1331    e. wcel 1480    o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-ov 5777  df-1st 6038
This theorem is referenced by:  ialgr0  11736  algrp1  11738
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