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Theorem algrflem 5773
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 5439 . 2 (B(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩)
2 fo1st 5707 . . . 4 1st :V–onto→V
3 fof 5031 . . . 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 3938 . . . 4 ((B V 𝐶 V) → ⟨B, 𝐶 V)
85, 6, 7mp2an 404 . . 3 B, 𝐶 V
9 fvco3 5169 . . 3 ((1st :V⟶V B, 𝐶 V) → ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩) = (𝐹‘(1st ‘⟨B, 𝐶⟩)))
104, 8, 9mp2an 404 . 2 ((𝐹 ∘ 1st )‘⟨B, 𝐶⟩) = (𝐹‘(1st ‘⟨B, 𝐶⟩))
115, 6op1st 5696 . . 3 (1st ‘⟨B, 𝐶⟩) = B
1211fveq2i 5106 . 2 (𝐹‘(1st ‘⟨B, 𝐶⟩)) = (𝐹B)
131, 10, 123eqtri 2046 1 (B(𝐹 ∘ 1st )𝐶) = (𝐹B)
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353  ccom 4276  wf 4825  ontowfo 4827  cfv 4829  (class class class)co 5436  1st c1st 5688
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-ov 5439  df-1st 5690
This theorem is referenced by: (None)
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