Theorem sbthlem7 9077

Description: Lemma for sbth 9081. (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 6578	. . 3 ⊢ (Fun 𝑓 → Fun (𝑓 ↾ ∪ 𝐷))
2		funres 6578	. . 3 ⊢ (Fun ◡𝑔 → Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmres 6011	. . . . . . . . 9 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
4		inss1 4189	. . . . . . . . 9 ⊢ (∪ 𝐷 ∩ dom 𝑓) ⊆ ∪ 𝐷
5	3, 4	eqsstri 3983	. . . . . . . 8 ⊢ dom (𝑓 ↾ ∪ 𝐷) ⊆ ∪ 𝐷
6		ssrin 4194	. . . . . . . 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 6011	. . . . . . . . 9 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
9		inss1 4189	. . . . . . . . 9 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) ⊆ (𝐴 ∖ ∪ 𝐷)
10	8, 9	eqsstri 3983	. . . . . . . 8 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ⊆ (𝐴 ∖ ∪ 𝐷)
11		sslin 4195	. . . . . . . 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 3946	. . . . . 6 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷))
14		disjdif 4433	. . . . . 6 ⊢ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷)) = ∅
15	13, 14	sseqtri 3985	. . . . 5 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ ∅
16		ss0 4359	. . . . 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 6582	. . . 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 703	. . 3 ⊢ ((Fun (𝑓 ↾ ∪ 𝐷) ∧ Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
20	1, 2, 19	syl2an 607	. 2 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
21		sbthlem.3	. . 3 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
22	21	funeqi 6557	. 2 ⊢ (Fun 𝐻 ↔ Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
23	20, 22	sylibr 237	1 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4872  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663   “ cima 5664  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-fun 6538

This theorem is referenced by:  sbthlem9  9079