Theorem sbthlem8 9151

Description: Lemma for sbth 9154. (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 6647	. . . 4 ⊢ (Fun ◡𝑓 → Fun ◡(𝑓 ↾ ∪ 𝐷))
2		funcnvcnv 6637	. . . . . 6 ⊢ (Fun 𝑔 → Fun ◡◡𝑔)
3		funres11 6647	. . . . . 6 ⊢ (Fun ◡◡𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4	2, 3	syl 18	. . . . 5 ⊢ (Fun 𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
5	4	ad3antrrr 743	. . . 4 ⊢ ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
6	1, 5	anim12i 625	. . 3 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun ◡(𝑓 ↾ ∪ 𝐷) ∧ Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
7		df-ima 5671	. . . . . . . 8 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
8		df-rn 5669	. . . . . . . 8 ⊢ ran (𝑓 ↾ ∪ 𝐷) = dom ◡(𝑓 ↾ ∪ 𝐷)
9	7, 8	eqtr2i 2787	. . . . . . 7 ⊢ dom ◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷)
10		df-ima 5671	. . . . . . . . 9 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
11		df-rn 5669	. . . . . . . . 9 ⊢ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
12	10, 11	eqtri 2786	. . . . . . . 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 9147	. . . . . . . 8 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
16	12, 15	eqtr3id 2812	. . . . . . 7 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
17		ineq12 4115	. . . . . . 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 599	. . . . . 6 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
19		disjdif 4429	. . . . . 6 ⊢ ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ∅
20	18, 19	eqtrdi 2814	. . . . 5 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
21	20	adantlll 731	. . . 4 ⊢ ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
22	21	adantl 487	. . 3 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
23		funun 6614	. . 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 596	. 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 5864	. . . 4 ⊢ ◡𝐻 = ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
27		cnvun 6151	. . . 4 ⊢ ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
28	26, 27	eqtri 2786	. . 3 ⊢ ◡𝐻 = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
29	28	funeqi 6588	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ∪ cuni 4872  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  Fun wfun 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568

This theorem is used by:  sbthlem9  9152