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 19483 | . . . 4
⊢ (𝐹 ∈ 𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷–1-1-onto→𝐷 ∧ dom (𝐹 ∖ I ) ≈
2o)) | 
| 4 | 3 | simp1bi 1146 | . . 3
⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V) | 
| 5 | 4 | adantr 480 | . 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐷 ∈ V) | 
| 6 |  | simpr 484 | . . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1-onto→𝐷) | 
| 7 | 1, 2 | pmtrff1o 19481 | . . . . 5
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) | 
| 8 | 7 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷–1-1-onto→𝐷) | 
| 9 |  | f1oco 6871 | . . . 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 584 | . . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷) | 
| 11 |  | f1ocnv 6860 | . . . 4
⊢ (𝐺:𝐷–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐷) | 
| 12 | 11 | adantl 481 | . . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ◡𝐺:𝐷–1-1-onto→𝐷) | 
| 13 |  | f1oco 6871 | . . 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 584 | . 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷) | 
| 15 |  | f1of 6848 | . . . . . . 7
⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷) | 
| 16 | 7, 15 | syl 17 | . . . . . 6
⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) | 
| 17 | 16 | adantr 480 | . . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷⟶𝐷) | 
| 18 |  | f1omvdconj 19464 | . . . . 5
⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) | 
| 19 | 17, 6, 18 | syl2anc 584 | . . . 4
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I ))) | 
| 20 |  | f1of1 6847 | . . . . . 6
⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷–1-1→𝐷) | 
| 21 | 20 | adantl 481 | . . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1→𝐷) | 
| 22 |  | difss 4136 | . . . . . . 7
⊢ (𝐹 ∖ I ) ⊆ 𝐹 | 
| 23 |  | dmss 5913 | . . . . . . 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 6755 | . . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ⊆ 𝐷) | 
| 26 | 5, 25 | ssexd 5324 | . . . . 5
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ∈ V) | 
| 27 |  | f1imaeng 9054 | . . . . 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 5165 | . . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I )) | 
| 30 | 3 | simp3bi 1148 | . . . 4
⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈
2o) | 
| 31 | 30 | adantr 480 | . . 3
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ≈
2o) | 
| 32 |  | entr 9046 | . . 3
⊢ ((dom
(((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) →
dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2o) | 
| 33 | 29, 31, 32 | syl2anc 584 | . 2
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈
2o) | 
| 34 | 1, 2 | pmtrfb 19483 | . 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 1344 | 1
⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅) |