Theorem sbthlem7 6851

Description: Lemma for isbth 6855. (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 5164	. . 3 ⊢ (Fun 𝑓 → Fun (𝑓 ↾ ∪ 𝐷))
2		funres 5164	. . 3 ⊢ (Fun ◡𝑔 → Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmres 4840	. . . . . . . . 9 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
4		inss1 3296	. . . . . . . . 9 ⊢ (∪ 𝐷 ∩ dom 𝑓) ⊆ ∪ 𝐷
5	3, 4	eqsstri 3129	. . . . . . . 8 ⊢ dom (𝑓 ↾ ∪ 𝐷) ⊆ ∪ 𝐷
6		ssrin 3301	. . . . . . . 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 4840	. . . . . . . . 9 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
9		inss1 3296	. . . . . . . . 9 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) ⊆ (𝐴 ∖ ∪ 𝐷)
10	8, 9	eqsstri 3129	. . . . . . . 8 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ⊆ (𝐴 ∖ ∪ 𝐷)
11		sslin 3302	. . . . . . . 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 3106	. . . . . 6 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷))
14		disjdif 3435	. . . . . 6 ⊢ (∪ 𝐷 ∩ (𝐴 ∖ ∪ 𝐷)) = ∅
15	13, 14	sseqtri 3131	. . . . 5 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∩ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⊆ ∅
16		ss0 3403	. . . . 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 5167	. . . 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 421	. . 3 ⊢ ((Fun (𝑓 ↾ ∪ 𝐷) ∧ Fun (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
20	1, 2, 19	syl2an 287	. 2 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
21		sbthlem.3	. . 3 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
22	21	funeqi 5144	. 2 ⊢ (Fun 𝐻 ↔ Fun ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
23	20, 22	sylibr 133	1 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2125  Vcvv 2686   ∖ cdif 3068   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  ∪ cuni 3736  ^◡ccnv 4538  dom cdm 4539   ↾ cres 4541   “ cima 4542  Fun wfun 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-fun 5125

This theorem is referenced by:  sbthlemi9  6853