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Theorem algrflemg 5851
 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 5515 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 5784 . . . . 5 1st :V–onto→V
3 fof 5106 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 7 . . . 4 1st :V⟶V
5 opexg 3964 . . . 4 ((𝐵𝑉𝐶𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6 fvco3 5244 . . . 4 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6sylancr 393 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
8 op1stg 5777 . . . 4 ((𝐵𝑉𝐶𝑊) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
98fveq2d 5182 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵))
107, 9eqtrd 2072 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹𝐵))
111, 10syl5eq 2084 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   ∘ ccom 4349  ⟶wf 4898  –onto→wfo 4900  ‘cfv 4902  (class class class)co 5512  1st c1st 5765 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-ov 5515  df-1st 5767 This theorem is referenced by:  ialgrlem1st  9881  ialgrp1  9885
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