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Theorem algrflemg 6136
 Description: Lemma for algrf 11782 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised 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 5786 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 6064 . . . . 5 1st :V–onto→V
3 fof 5354 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . . 4 1st :V⟶V
5 opexg 4159 . . . 4 ((𝐵𝑉𝐶𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6 fvco3 5501 . . . 4 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6sylancr 411 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
8 op1stg 6057 . . . 4 ((𝐵𝑉𝐶𝑊) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
98fveq2d 5434 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵))
107, 9eqtrd 2173 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹𝐵))
111, 10syl5eq 2185 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  Vcvv 2690  ⟨cop 3536   ∘ ccom 4552  ⟶wf 5128  –onto→wfo 5130  ‘cfv 5132  (class class class)co 5783  1st c1st 6045 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fo 5138  df-fv 5140  df-ov 5786  df-1st 6047 This theorem is referenced by:  ialgrlem1st  11779  algrp1  11783
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