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Theorem algrflemg 5977
Description: Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
Assertion
Ref Expression
algrflemg ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))

Proof of Theorem algrflemg
StepHypRef Expression
1 df-ov 5637 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 5910 . . . . 5 1st :V–onto→V
3 fof 5217 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 7 . . . 4 1st :V⟶V
5 opexg 4046 . . . 4 ((𝐵𝑉𝐶𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6 fvco3 5359 . . . 4 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6sylancr 405 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
8 op1stg 5903 . . . 4 ((𝐵𝑉𝐶𝑊) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
98fveq2d 5293 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵))
107, 9eqtrd 2120 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹𝐵))
111, 10syl5eq 2132 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  cop 3444  ccom 4432  wf 4998  ontowfo 5000  cfv 5002  (class class class)co 5634  1st c1st 5891
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-1st 5893
This theorem is referenced by:  ialgrlem1st  11106  ialgrp1  11110
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