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Theorem algrflem 7806
 Description: Lemma for algrf 15910 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 7142 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 7695 . . . 4 1st :V–onto→V
3 fof 6569 . . . 4 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . 3 1st :V⟶V
5 opex 5324 . . 3 𝐵, 𝐶⟩ ∈ V
6 fvco3 6741 . . 3 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6mp2an 691 . 2 ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩))
8 algrflem.1 . . . 4 𝐵 ∈ V
9 algrflem.2 . . . 4 𝐶 ∈ V
108, 9op1st 7683 . . 3 (1st ‘⟨𝐵, 𝐶⟩) = 𝐵
1110fveq2i 6652 . 2 (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵)
121, 7, 113eqtri 2828 1 (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⟨cop 4534   ∘ ccom 5527  ⟶wf 6324  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7139  1st c1st 7673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-ov 7142  df-1st 7675 This theorem is referenced by:  fpwwe  10061  seq1st  15908  algrf  15910  algrp1  15911  dvnff  24529  dvnp1  24531
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