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Theorem algrflem 7173
Description: Lemma for algrf 15124 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 6552 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 7079 . . . 4 1st :V–onto→V
3 fof 6028 . . . 4 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . 3 1st :V⟶V
5 opex 4859 . . 3 𝐵, 𝐶⟩ ∈ V
6 fvco3 6185 . . 3 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6mp2an 704 . 2 ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩))
8 algrflem.1 . . . 4 𝐵 ∈ V
9 algrflem.2 . . . 4 𝐶 ∈ V
108, 9op1st 7067 . . 3 (1st ‘⟨𝐵, 𝐶⟩) = 𝐵
1110fveq2i 6106 . 2 (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵)
121, 7, 113eqtri 2636 1 (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  Vcvv 3173  cop 4131  ccom 5042  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  1st c1st 7057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-1st 7059
This theorem is referenced by:  fpwwe  9347  seq1st  15122  algrf  15124  algrp1  15125  dvnff  23492  dvnp1  23494
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