Theorem sbthlem8 9130

Description: Lemma for sbth 9133. (Contributed by NM, 27-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
sbthlem.3	⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))

Assertion

Ref	Expression
sbthlem8	⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔   𝑥,𝐻

Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

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)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563

This theorem is referenced by:  sbthlem9  9131