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 18890 |
. . . 4
⊢ (𝐹 ∈ 𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷–1-1-onto→𝐷 ∧ dom (𝐹 ∖ I ) ≈
2o)) |
4 | 3 | simp1bi 1147 |
. . 3
⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V) |
5 | 4 | adantr 484 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐷 ∈ V) |
6 | | simpr 488 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1-onto→𝐷) |
7 | 1, 2 | pmtrff1o 18888 |
. . . . 5
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) |
8 | 7 | adantr 484 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷–1-1-onto→𝐷) |
9 | | f1oco 6705 |
. . . 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 587 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷) |
11 | | f1ocnv 6695 |
. . . 4
⊢ (𝐺:𝐷–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐷) |
12 | 11 | adantl 485 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ◡𝐺:𝐷–1-1-onto→𝐷) |
13 | | f1oco 6705 |
. . 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 587 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷) |
15 | | f1of 6683 |
. . . . . . 7
⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷) |
16 | 7, 15 | syl 17 |
. . . . . 6
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) |
17 | 16 | adantr 484 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷⟶𝐷) |
18 | | f1omvdconj 18871 |
. . . . 5
⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) |
19 | 17, 6, 18 | syl2anc 587 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) |
20 | | f1of1 6682 |
. . . . . 6
⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷–1-1→𝐷) |
21 | 20 | adantl 485 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1→𝐷) |
22 | | difss 4063 |
. . . . . . 7
⊢ (𝐹 ∖ I ) ⊆ 𝐹 |
23 | | dmss 5789 |
. . . . . . 7
⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹) |
24 | 22, 23 | ax-mp 5 |
. . . . . 6
⊢ dom
(𝐹 ∖ I ) ⊆ dom
𝐹 |
25 | 24, 17 | fssdm 6587 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ⊆ 𝐷) |
26 | 5, 25 | ssexd 5234 |
. . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ∈ V) |
27 | | f1imaeng 8714 |
. . . . 5
⊢ ((𝐺:𝐷–1-1→𝐷 ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ∈ V) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I )) |
28 | 21, 25, 26, 27 | syl3anc 1373 |
. . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I )) |
29 | 19, 28 | eqbrtrd 5092 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I )) |
30 | 3 | simp3bi 1149 |
. . . 4
⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈
2o) |
31 | 30 | adantr 484 |
. . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ≈
2o) |
32 | | entr 8706 |
. . 3
⊢ ((dom
(((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) →
dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2o) |
33 | 29, 31, 32 | syl2anc 587 |
. 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2o) |
34 | 1, 2 | pmtrfb 18890 |
. 2
⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅 ↔ (𝐷 ∈ V ∧ ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷 ∧ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2o)) |
35 | 5, 14, 33, 34 | syl3anbrc 1345 |
1
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅) |