Theorem ialgrlem1st 11974

Description: Lemma for ialgr0 11976. Expressing algrflemg 6198 in a form suitable for theorems such as seq3-1 10395 or seqf 10396. (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

Step	Hyp	Ref	Expression
1		algrflemg 6198	. . 3 |- ( ( x e. S /\ y e. S ) -> ( x ( F o. 1st ) y ) = ( F ` x ) )
2	1	adantl 275	. 2 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) = ( F ` x ) )
3		ialgrlem1st.f	. . . 4 |- ( ph -> F : S --> S )
4	3	adantr 274	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> F : S --> S )
5		simprl 521	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> x e. S )
6	4, 5	ffvelrnd 5621	. 2 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
7	2, 6	eqeltrd 2243	1 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   o. ccom 4608   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-ov 5845  df-1st 6108

This theorem is referenced by:  ialgr0  11976  algrp1  11978