Theorem ialgrlem1st 11996

Description: Lemma for ialgr0 11998. Expressing algrflemg 6209 in a form suitable for theorems such as seq3-1 10416 or seqf 10417. (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 6209	. . 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 526	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> x e. S )
6	4, 5	ffvelrnd 5632	. 2 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
7	2, 6	eqeltrd 2247	1 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   o. ccom 4615   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-ov 5856  df-1st 6119

This theorem is referenced by:  ialgr0  11998  algrp1  12000