Proof of Theorem pmtrfconj
Step | Hyp | Ref
| Expression |
1 | | pmtrrn.t |
. . . . 5
⊢ 𝑇 = (pmTrsp‘𝐷) |
2 | | pmtrrn.r |
. . . . 5
⊢ 𝑅 = ran 𝑇 |
3 | 1, 2 | pmtrfb 17885 |
. . . 4
⊢ (𝐹 ∈ 𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷–1-1-onto→𝐷 ∧ dom (𝐹 ∖ I ) ≈
2𝑜)) |
4 | 3 | simp1bi 1076 |
. . 3
⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V) |
5 | 4 | adantr 481 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐷 ∈ V) |
6 | | simpr 477 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1-onto→𝐷) |
7 | 1, 2 | pmtrff1o 17883 |
. . . . 5
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) |
8 | 7 | adantr 481 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷–1-1-onto→𝐷) |
9 | | f1oco 6159 |
. . . 4
⊢ ((𝐺:𝐷–1-1-onto→𝐷 ∧ 𝐹:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷) |
10 | 6, 8, 9 | syl2anc 693 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷) |
11 | | f1ocnv 6149 |
. . . 4
⊢ (𝐺:𝐷–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐷) |
12 | 11 | adantl 482 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ◡𝐺:𝐷–1-1-onto→𝐷) |
13 | | f1oco 6159 |
. . 3
⊢ (((𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷 ∧ ◡𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷) |
14 | 10, 12, 13 | syl2anc 693 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷) |
15 | | f1of 6137 |
. . . . . . 7
⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷) |
16 | 7, 15 | syl 17 |
. . . . . 6
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) |
17 | 16 | adantr 481 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷⟶𝐷) |
18 | | f1omvdconj 17866 |
. . . . 5
⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) |
19 | 17, 6, 18 | syl2anc 693 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) |
20 | | f1of1 6136 |
. . . . . 6
⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷–1-1→𝐷) |
21 | 20 | adantl 482 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1→𝐷) |
22 | | difss 3737 |
. . . . . . . 8
⊢ (𝐹 ∖ I ) ⊆ 𝐹 |
23 | | dmss 5323 |
. . . . . . . 8
⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹) |
24 | 22, 23 | ax-mp 5 |
. . . . . . 7
⊢ dom
(𝐹 ∖ I ) ⊆ dom
𝐹 |
25 | | fdm 6051 |
. . . . . . 7
⊢ (𝐹:𝐷⟶𝐷 → dom 𝐹 = 𝐷) |
26 | 24, 25 | syl5sseq 3653 |
. . . . . 6
⊢ (𝐹:𝐷⟶𝐷 → dom (𝐹 ∖ I ) ⊆ 𝐷) |
27 | 17, 26 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ⊆ 𝐷) |
28 | 5, 27 | ssexd 4805 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ∈ V) |
29 | | f1imaeng 8016 |
. . . . 5
⊢ ((𝐺:𝐷–1-1→𝐷 ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ∈ V) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I )) |
30 | 21, 27, 28, 29 | syl3anc 1326 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I )) |
31 | 19, 30 | eqbrtrd 4675 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I )) |
32 | 3 | simp3bi 1078 |
. . . 4
⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈
2𝑜) |
33 | 32 | adantr 481 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ≈
2𝑜) |
34 | | entr 8008 |
. . 3
⊢ ((dom
(((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2𝑜)
→ dom (((𝐺 ∘
𝐹) ∘ ◡𝐺) ∖ I ) ≈
2𝑜) |
35 | 31, 33, 34 | syl2anc 693 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2𝑜) |
36 | 1, 2 | pmtrfb 17885 |
. 2
⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅 ↔ (𝐷 ∈ V ∧ ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷 ∧ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2𝑜)) |
37 | 5, 14, 35, 36 | syl3anbrc 1246 |
1
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅) |