Theorem sbthlem7 9155

Description: Lemma for sbth 9159. (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
sbthlem7	⊢ ((Fun 𝑓 ∧ 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 sbthlem7

Step	Hyp	Ref	Expression
1		funres 6620	. . 3 ⊢ (Fun 𝑓 → Fun (𝑓 ↾ ∪ 𝐷))
2		funres 6620	. . 3 ⊢ (Fun ◡𝑔 → Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmres 6041	. . . . . . . . 9 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
4		inss1 4258	. . . . . . . . 9 ⊢ (∪ 𝐷 ∩ dom 𝑓) ⊆ ∪ 𝐷
5	3, 4	eqsstri 4043	. . . . . . . 8 ⊢ dom (𝑓 ↾ ∪ 𝐷) ⊆ ∪ 𝐷
6		ssrin 4263	. . . . . . . 8 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ⊆ ∪ 𝐷 → (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		dmres 6041	. . . . . . . . 9 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
9		inss1 4258	. . . . . . . . 9 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) ⊆ (𝐴 ∖ ∪ 𝐷)
10	8, 9	eqsstri 4043	. . . . . . . 8 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ⊆ (𝐴 ∖ ∪ 𝐷)
11		sslin 4264	. . . . . . . 8 ⊢ (dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ⊆ (𝐴 ∖ ∪ 𝐷) → (∪ 𝐷 ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷)))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (∪ 𝐷 ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷))
13	7, 12	sstri 4018	. . . . . 6 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷))
14		disjdif 4495	. . . . . 6 ⊢ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷)) = ∅
15	13, 14	sseqtri 4045	. . . . 5 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ ∅
16		ss0 4425	. . . . 5 ⊢ ((dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ ∅ → (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
17	15, 16	ax-mp 5	. . . 4 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅
18		funun 6624	. . . 4 ⊢ (((Fun (𝑓 ↾ ∪ 𝐷) ∧ Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∧ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
19	17, 18	mpan2 690	. . 3 ⊢ ((Fun (𝑓 ↾ ∪ 𝐷) ∧ Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
20	1, 2, 19	syl2an 595	. 2 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
21		sbthlem.3	. . 3 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
22	21	funeqi 6599	. 2 ⊢ (Fun 𝐻 ↔ Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
23	20, 22	sylibr 234	1 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702   “ cima 5703  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-fun 6575

This theorem is referenced by:  sbthlem9  9157