Theorem sbthlem8 9082

Description: Lemma for sbth 9085. (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 6614	. . . 4 ⊢ (Fun ◡𝑓 → Fun ◡(𝑓 ↾ ∪ 𝐷))
2		funcnvcnv 6604	. . . . . 6 ⊢ (Fun 𝑔 → Fun ◡◡𝑔)
3		funres11 6614	. . . . . 6 ⊢ (Fun ◡◡𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4	2, 3	syl 18	. . . . 5 ⊢ (Fun 𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
5	4	ad3antrrr 742	. . . 4 ⊢ ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
6	1, 5	anim12i 624	. . 3 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun ◡(𝑓 ↾ ∪ 𝐷) ∧ Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
7		df-ima 5675	. . . . . . . 8 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
8		df-rn 5673	. . . . . . . 8 ⊢ ran (𝑓 ↾ ∪ 𝐷) = dom ◡(𝑓 ↾ ∪ 𝐷)
9	7, 8	eqtr2i 2793	. . . . . . 7 ⊢ dom ◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷)
10		df-ima 5675	. . . . . . . . 9 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
11		df-rn 5673	. . . . . . . . 9 ⊢ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
12	10, 11	eqtri 2792	. . . . . . . 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 9078	. . . . . . . 8 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
16	12, 15	eqtr3id 2818	. . . . . . 7 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
17		ineq12 4176	. . . . . . 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 598	. . . . . 6 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
19		disjdif 4438	. . . . . 6 ⊢ ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ∅
20	18, 19	eqtrdi 2820	. . . . 5 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
21	20	adantlll 730	. . . 4 ⊢ ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
22	21	adantl 486	. . 3 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
23		funun 6583	. . 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 595	. 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 5861	. . . 4 ⊢ ◡𝐻 = ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
27		cnvun 6140	. . . 4 ⊢ ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
28	26, 27	eqtri 2792	. . 3 ⊢ ◡𝐻 = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
29	28	funeqi 6558	. 2 ⊢ (Fun ◡𝐻 ↔ Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
30	24, 29	sylibr 237	1 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4876  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   ↾ cres 5664   “ cima 5665  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-fun 6539

This theorem is referenced by:  sbthlem9  9083