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Theorem ialgrlem1st 10568
Description: Lemma for ialgr0 10570. Expressing algrflemg 5882 in a form suitable for theorems such as iseq1 9533 or iseqfcl 9535. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypothesis
Ref Expression
ialgrlem1st.f (𝜑𝐹:𝑆𝑆)
Assertion
Ref Expression
ialgrlem1st ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

Proof of Theorem ialgrlem1st
StepHypRef Expression
1 algrflemg 5882 . . 3 ((𝑥𝑆𝑦𝑆) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
21adantl 271 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
3 ialgrlem1st.f . . . 4 (𝜑𝐹:𝑆𝑆)
43adantr 270 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝐹:𝑆𝑆)
5 simprl 498 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
64, 5ffvelrnd 5335 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹𝑥) ∈ 𝑆)
72, 6eqeltrd 2156 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  ccom 4375  wf 4928  cfv 4932  (class class class)co 5543  1st c1st 5796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-ov 5546  df-1st 5798
This theorem is referenced by:  ialgr0  10570  ialgrf  10571  ialgrp1  10572
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