Theorem ialgrlem1st 12585

Description: Lemma for ialgr0 12587. Expressing algrflemg 6387 in a form suitable for theorems such as seq3-1 10701 or seqf 10703. (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 6387	. . 3 |- ( ( x e. S /\ y e. S ) -> ( x ( F o. 1st ) y ) = ( F ` x ) )
2	1	adantl 277	. 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 276	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> F : S --> S )
5		simprl 529	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> x e. S )
6	4, 5	ffvelcdmd 5776	. 2 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
7	2, 6	eqeltrd 2306	1 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   o. ccom 4724   -->wf 5317   ` cfv 5321  (class class class)co 6010   1stc1st 6293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fo 5327  df-fv 5329  df-ov 6013  df-1st 6295

This theorem is referenced by:  ialgr0  12587  algrp1  12589