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Theorem algrflem 5878
 Description: Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
algrflem.1 𝐵 ∈ V
algrflem.2 𝐶 ∈ V
Assertion
Ref Expression
algrflem (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)

Proof of Theorem algrflem
StepHypRef Expression
1 df-ov 5543 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 5812 . . . 4 1st :V–onto→V
3 fof 5134 . . . 4 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 7 . . 3 1st :V⟶V
5 algrflem.1 . . . 4 𝐵 ∈ V
6 algrflem.2 . . . 4 𝐶 ∈ V
7 opexg 3992 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐵, 𝐶⟩ ∈ V)
85, 6, 7mp2an 410 . . 3 𝐵, 𝐶⟩ ∈ V
9 fvco3 5272 . . 3 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
104, 8, 9mp2an 410 . 2 ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩))
115, 6op1st 5801 . . 3 (1st ‘⟨𝐵, 𝐶⟩) = 𝐵
1211fveq2i 5209 . 2 (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵)
131, 10, 123eqtri 2080 1 (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1259   ∈ wcel 1409  Vcvv 2574  ⟨cop 3406   ∘ ccom 4377  ⟶wf 4926  –onto→wfo 4928  ‘cfv 4930  (class class class)co 5540  1st c1st 5793 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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-ov 5543  df-1st 5795 This theorem is referenced by: (None)
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