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Theorem algrflemg 6225
Description: Lemma for algrf 12025 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 5872 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 6152 . . . . 5 1st :V–onto→V
3 fof 5434 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . . 4 1st :V⟶V
5 opexg 4225 . . . 4 ((𝐵𝑉𝐶𝑊) → ⟨𝐵, 𝐶⟩ ∈ V)
6 fvco3 5583 . . . 4 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6sylancr 414 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
8 op1stg 6145 . . . 4 ((𝐵𝑉𝐶𝑊) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
98fveq2d 5515 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵))
107, 9eqtrd 2210 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹𝐵))
111, 10eqtrid 2222 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cop 3594  ccom 4627  wf 5208  ontowfo 5210  cfv 5212  (class class class)co 5869  1st c1st 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220  df-ov 5872  df-1st 6135
This theorem is referenced by:  ialgrlem1st  12022  algrp1  12026
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