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Theorem algrflem 5792
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 5458 . 2 (B(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩)
2 fo1st 5726 . . . 4 1st :V–onto→V
3 fof 5049 . . . 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 3955 . . . 4 ((B V 𝐶 V) → ⟨B, 𝐶 V)
85, 6, 7mp2an 402 . . 3 B, 𝐶 V
9 fvco3 5187 . . 3 ((1st :V⟶V B, 𝐶 V) → ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩) = (𝐹‘(1st ‘⟨B, 𝐶⟩)))
104, 8, 9mp2an 402 . 2 ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩) = (𝐹‘(1st ‘⟨B, 𝐶⟩))
115, 6op1st 5715 . . 3 (1st ‘⟨B, 𝐶⟩) = B
1211fveq2i 5124 . 2 (𝐹‘(1st ‘⟨B, 𝐶⟩)) = (𝐹B)
131, 10, 123eqtri 2061 1 (B(𝐹 ∘ 1st )𝐶) = (𝐹B)
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  cop 3370  ccom 4292  wf 4841  ontowfo 4843  cfv 4845  (class class class)co 5455  1st c1st 5707
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-ov 5458  df-1st 5709
This theorem is referenced by: (None)
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