Proof of Theorem sbthlem8
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | funres11 6643 | . . . 4
⊢ (Fun
◡𝑓 → Fun ◡(𝑓 ↾ ∪ 𝐷)) | 
| 2 |  | funcnvcnv 6633 | . . . . . 6
⊢ (Fun
𝑔 → Fun ◡◡𝑔) | 
| 3 |  | funres11 6643 | . . . . . 6
⊢ (Fun
◡◡𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 4 | 2, 3 | syl 17 | . . . . 5
⊢ (Fun
𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 5 | 4 | ad3antrrr 730 | . . . 4
⊢ ((((Fun
𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 6 | 1, 5 | anim12i 613 | . . 3
⊢ ((Fun
◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun ◡(𝑓 ↾ ∪ 𝐷) ∧ Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))) | 
| 7 |  | df-ima 5698 | . . . . . . . 8
⊢ (𝑓 “ ∪ 𝐷) =
ran (𝑓 ↾ ∪ 𝐷) | 
| 8 |  | df-rn 5696 | . . . . . . . 8
⊢ ran
(𝑓 ↾ ∪ 𝐷) =
dom ◡(𝑓 ↾ ∪ 𝐷) | 
| 9 | 7, 8 | eqtr2i 2766 | . . . . . . 7
⊢ dom ◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷) | 
| 10 |  | df-ima 5698 | . . . . . . . . 9
⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) | 
| 11 |  | df-rn 5696 | . . . . . . . . 9
⊢ ran
(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) | 
| 12 | 10, 11 | eqtri 2765 | . . . . . . . 8
⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) | 
| 13 |  | sbthlem.1 | . . . . . . . . 9
⊢ 𝐴 ∈ V | 
| 14 |  | sbthlem.2 | . . . . . . . . 9
⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | 
| 15 | 13, 14 | sbthlem4 9126 | . . . . . . . 8
⊢ (((dom
𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) | 
| 16 | 12, 15 | eqtr3id 2791 | . . . . . . 7
⊢ (((dom
𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) | 
| 17 |  | ineq12 4215 | . . . . . . 7
⊢ ((dom
◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷) ∧ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) | 
| 18 | 9, 16, 17 | sylancr 587 | . . . . . 6
⊢ (((dom
𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) | 
| 19 |  | disjdif 4472 | . . . . . 6
⊢ ((𝑓 “ ∪ 𝐷)
∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
= ∅ | 
| 20 | 18, 19 | eqtrdi 2793 | . . . . 5
⊢ (((dom
𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) | 
| 21 | 20 | adantlll 718 | . . . 4
⊢ ((((Fun
𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) | 
| 22 | 21 | adantl 481 | . . 3
⊢ ((Fun
◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) | 
| 23 |  | funun 6612 | . . 3
⊢ (((Fun
◡(𝑓 ↾ ∪ 𝐷) ∧ Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∧ (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) → Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))) | 
| 24 | 6, 22, 23 | syl2anc 584 | . 2
⊢ ((Fun
◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))) | 
| 25 |  | sbthlem.3 | . . . . 5
⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 26 | 25 | cnveqi 5885 | . . . 4
⊢ ◡𝐻 = ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 27 |  | cnvun 6162 | . . . 4
⊢ ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 28 | 26, 27 | eqtri 2765 | . . 3
⊢ ◡𝐻 = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | 
| 29 | 28 | funeqi 6587 | . 2
⊢ (Fun
◡𝐻 ↔ Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))) | 
| 30 | 24, 29 | sylibr 234 | 1
⊢ ((Fun
◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻) |