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Theorem algrflemg 5876
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 5540 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 5809 . . . . 5 1st :V–onto→V
3 fof 5131 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 7 . . . 4 1st :V⟶V
5 opexg 3989 . . . 4 ((𝐵𝑉𝐶𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6 fvco3 5269 . . . 4 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6sylancr 399 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
8 op1stg 5802 . . . 4 ((𝐵𝑉𝐶𝑊) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
98fveq2d 5207 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵))
107, 9eqtrd 2086 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹𝐵))
111, 10syl5eq 2098 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wcel 1407  Vcvv 2572  cop 3403  ccom 4374  wf 4923  ontowfo 4925  cfv 4927  (class class class)co 5537  1st c1st 5790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-fo 4933  df-fv 4935  df-ov 5540  df-1st 5792
This theorem is referenced by:  ialgrlem1st  10207  ialgrp1  10211
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