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| Mirrors > Home > MPE Home > Th. List > efgmf | Structured version Visualization version GIF version | ||
| Description: The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
| Ref | Expression |
|---|---|
| efgmval.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
| Ref | Expression |
|---|---|
| efgmf | ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2oconcl 8541 | . . . 4 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o) | |
| 2 | opelxpi 5722 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) | |
| 3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) |
| 4 | 3 | rgen2 3199 | . 2 ⊢ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) |
| 5 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
| 6 | 5 | fmpo 8093 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) |
| 7 | 4, 6 | mpbi 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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