Proof of Theorem sbthlemi4
Step | Hyp | Ref
| Expression |
1 | | df-ima 4625 |
. 2
⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) |
2 | | difss 3254 |
. . . . . . . 8
⊢ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵 |
3 | | sseq2 3172 |
. . . . . . . 8
⊢ (dom
𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵)) |
4 | 2, 3 | mpbiri 167 |
. . . . . . 7
⊢ (dom
𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔) |
5 | | ssdmres 4914 |
. . . . . . 7
⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) |
6 | 4, 5 | sylib 121 |
. . . . . 6
⊢ (dom
𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) |
7 | | dfdm4 4804 |
. . . . . 6
⊢ dom
(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) |
8 | 6, 7 | eqtr3di 2219 |
. . . . 5
⊢ (dom
𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) |
9 | 8 | adantr 274 |
. . . 4
⊢ ((dom
𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) |
10 | 9 | 3ad2ant2 1015 |
. . 3
⊢
((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) |
11 | | funcnvres 5273 |
. . . . . . 7
⊢ (Fun
◡𝑔 → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) |
12 | 11 | 3ad2ant3 1016 |
. . . . . 6
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) |
13 | | sbthlem.1 |
. . . . . . . . 9
⊢ 𝐴 ∈ V |
14 | | sbthlem.2 |
. . . . . . . . 9
⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} |
15 | 13, 14 | sbthlemi3 6940 |
. . . . . . . 8
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷)) |
16 | 15 | reseq2d 4892 |
. . . . . . 7
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
17 | 16 | 3adant3 1013 |
. . . . . 6
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
18 | 12, 17 | eqtrd 2204 |
. . . . 5
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
19 | 18 | rneqd 4841 |
. . . 4
⊢
((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
20 | 19 | 3adant2l 1228 |
. . 3
⊢
((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
21 | 10, 20 | eqtrd 2204 |
. 2
⊢
((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) |
22 | 1, 21 | eqtr4id 2223 |
1
⊢
((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) |