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Theorem algrflem 7454
Description: Lemma for algrf 15488 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 6816 . 2 (𝐵(𝐹 ∘ 1st )𝐶) = ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩)
2 fo1st 7353 . . . 4 1st :V–onto→V
3 fof 6276 . . . 4 (1st :V–onto→V → 1st :V⟶V)
42, 3ax-mp 5 . . 3 1st :V⟶V
5 opex 5081 . . 3 𝐵, 𝐶⟩ ∈ V
6 fvco3 6437 . . 3 ((1st :V⟶V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)))
74, 5, 6mp2an 710 . 2 ((𝐹 ∘ 1st )‘⟨𝐵, 𝐶⟩) = (𝐹‘(1st ‘⟨𝐵, 𝐶⟩))
8 algrflem.1 . . . 4 𝐵 ∈ V
9 algrflem.2 . . . 4 𝐶 ∈ V
108, 9op1st 7341 . . 3 (1st ‘⟨𝐵, 𝐶⟩) = 𝐵
1110fveq2i 6355 . 2 (𝐹‘(1st ‘⟨𝐵, 𝐶⟩)) = (𝐹𝐵)
121, 7, 113eqtri 2786 1 (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327  ccom 5270  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  1st c1st 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-ov 6816  df-1st 7333
This theorem is referenced by:  fpwwe  9660  seq1st  15486  algrf  15488  algrp1  15489  dvnff  23885  dvnp1  23887
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