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Theorem efgmf 19731
Description: The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypothesis
Ref Expression
efgmval.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
Assertion
Ref Expression
efgmf 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
Distinct variable group:   𝑦,𝑧,𝐼
Allowed substitution hints:   𝑀(𝑦,𝑧)

Proof of Theorem efgmf
StepHypRef Expression
1 2oconcl 8541 . . . 4 (𝑧 ∈ 2o → (1o𝑧) ∈ 2o)
2 opelxpi 5722 . . . 4 ((𝑦𝐼 ∧ (1o𝑧) ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
31, 2sylan2 593 . . 3 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
43rgen2 3199 . 2 𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
65fmpo 8093 . 2 (∀𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
74, 6mpbi 230 1 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cop 4632   × cxp 5683  wf 6557  cmpo 7433  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507
This theorem is referenced by:  efgtf  19740  efgtlen  19744  efginvrel2  19745  efginvrel1  19746  efgredleme  19761  efgredlemc  19763  efgcpbllemb  19773  frgp0  19778  frgpinv  19782  vrgpinv  19787  frgpnabllem1  19891
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