Proof of Theorem resin
Step | Hyp | Ref
| Expression |
1 | | resdif 6737 |
. . . 4
⊢ ((Fun
◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷)) |
2 | | f1ofo 6723 |
. . . 4
⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷)) |
3 | 1, 2 | syl 17 |
. . 3
⊢ ((Fun
◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷)) |
4 | | resdif 6737 |
. . 3
⊢ ((Fun
◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–onto→(𝐶 ∖ 𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))) |
5 | 3, 4 | syld3an3 1408 |
. 2
⊢ ((Fun
◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))) |
6 | | dfin4 4201 |
. . . 4
⊢ (𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷)) |
7 | | f1oeq3 6706 |
. . . 4
⊢ ((𝐶 ∩ 𝐷) = (𝐶 ∖ (𝐶 ∖ 𝐷)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))) |
8 | 6, 7 | ax-mp 5 |
. . 3
⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))) |
9 | | dfin4 4201 |
. . . 4
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
10 | | f1oeq2 6705 |
. . . 4
⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))) |
11 | 9, 10 | ax-mp 5 |
. . 3
⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))) |
12 | 9 | reseq2i 5888 |
. . . 4
⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))) |
13 | | f1oeq1 6704 |
. . . 4
⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)))) |
14 | 12, 13 | ax-mp 5 |
. . 3
⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷))) |
15 | 8, 11, 14 | 3bitrri 298 |
. 2
⊢ ((𝐹 ↾ (𝐴 ∖ (𝐴 ∖ 𝐵))):(𝐴 ∖ (𝐴 ∖ 𝐵))–1-1-onto→(𝐶 ∖ (𝐶 ∖ 𝐷)) ↔ (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷)) |
16 | 5, 15 | sylib 217 |
1
⊢ ((Fun
◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷)) |