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Theorem efgmf 19495
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 8449 . . . 4 (𝑧 ∈ 2o → (1o𝑧) ∈ 2o)
2 opelxpi 5670 . . . 4 ((𝑦𝐼 ∧ (1o𝑧) ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
31, 2sylan2 593 . . 3 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
43rgen2 3194 . 2 𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
65fmpo 8000 . 2 (∀𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
74, 6mpbi 229 1 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wral 3064  cdif 3907  cop 4592   × cxp 5631  wf 6492  cmpo 7359  1oc1o 8405  2oc2o 8406
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-1o 8412  df-2o 8413
This theorem is referenced by:  efgtf  19504  efgtlen  19508  efginvrel2  19509  efginvrel1  19510  efgredleme  19525  efgredlemc  19527  efgcpbllemb  19537  frgp0  19542  frgpinv  19546  vrgpinv  19551  frgpnabllem1  19651
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