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Theorem sbthlem7 9072
Description: Lemma for sbth 9076. (Contributed by NM, 27-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
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
StepHypRef Expression
1 funres 6579 . . 3 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funres 6579 . . 3 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
3 dmres 5995 . . . . . . . . 9 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
4 inss1 4224 . . . . . . . . 9 ( 𝐷 ∩ dom 𝑓) ⊆ 𝐷
53, 4eqsstri 4012 . . . . . . . 8 dom (𝑓 𝐷) ⊆ 𝐷
6 ssrin 4229 . . . . . . . 8 (dom (𝑓 𝐷) ⊆ 𝐷 → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))))
75, 6ax-mp 5 . . . . . . 7 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷)))
8 dmres 5995 . . . . . . . . 9 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
9 inss1 4224 . . . . . . . . 9 ((𝐴 𝐷) ∩ dom 𝑔) ⊆ (𝐴 𝐷)
108, 9eqsstri 4012 . . . . . . . 8 dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷)
11 sslin 4230 . . . . . . . 8 (dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷) → ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷)))
1210, 11ax-mp 5 . . . . . . 7 ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
137, 12sstri 3987 . . . . . 6 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
14 disjdif 4467 . . . . . 6 ( 𝐷 ∩ (𝐴 𝐷)) = ∅
1513, 14sseqtri 4014 . . . . 5 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅
16 ss0 4394 . . . . 5 ((dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅ → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
1715, 16ax-mp 5 . . . 4 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅
18 funun 6583 . . . 4 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
1917, 18mpan2 689 . . 3 ((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
201, 2, 19syl2an 596 . 2 ((Fun 𝑓 ∧ Fun 𝑔) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
21 sbthlem.3 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2221funeqi 6558 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
2320, 22sylibr 233 1 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2708  Vcvv 3473  cdif 3941  cun 3942  cin 3943  wss 3944  c0 4318   cuni 4901  ccnv 5668  dom cdm 5669  cres 5671  cima 5672  Fun wfun 6526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-fun 6534
This theorem is referenced by:  sbthlem9  9074
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