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Theorem algrflem 6269
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 5909 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 6197 . . . 4 1st :V–onto→V
3 fof 5464 . . . 4 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . 3 1st :V⟶V
5 algrflem.1 . . . 4 𝐵 ∈ V
6 algrflem.2 . . . 4 𝐶 ∈ V
7 opexg 4253 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐵, 𝐶⟩ ∈ V)
85, 6, 7mp2an 426 . . 3 𝐵, 𝐶⟩ ∈ V
9 fvco3 5616 . . 3 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
104, 8, 9mp2an 426 . 2 ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩))
115, 6op1st 6186 . . 3 (1st ‘⟨𝐵, 𝐶⟩) = 𝐵
1211fveq2i 5545 . 2 (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵)
131, 10, 123eqtri 2214 1 (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  Vcvv 2756  cop 3617  ccom 4655  wf 5238  ontowfo 5240  cfv 5242  (class class class)co 5906  1st c1st 6178
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-sbc 2982  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fo 5248  df-fv 5250  df-ov 5909  df-1st 6180
This theorem is referenced by:  algrf  12157
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