Theorem ialgrlem1st 12009

Description: Lemma for ialgr0 12011. Expressing algrflemg 6221 in a form suitable for theorems such as seq3-1 10430 or seqf 10431. (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 6221	. . 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 5644	. 2 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
7	2, 6	eqeltrd 2252	1 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   o. ccom 4624   -->wf 5204   ` cfv 5208  (class class class)co 5865   1stc1st 6129

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fo 5214  df-fv 5216  df-ov 5868  df-1st 6131

This theorem is referenced by:  ialgr0  12011  algrp1  12013