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Theorem sbthlem7 9092
Description: Lemma for sbth 9096. (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 6591 . . 3 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funres 6591 . . 3 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
3 dmres 6004 . . . . . . . . 9 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
4 inss1 4229 . . . . . . . . 9 ( 𝐷 ∩ dom 𝑓) ⊆ 𝐷
53, 4eqsstri 4017 . . . . . . . 8 dom (𝑓 𝐷) ⊆ 𝐷
6 ssrin 4234 . . . . . . . 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 6004 . . . . . . . . 9 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
9 inss1 4229 . . . . . . . . 9 ((𝐴 𝐷) ∩ dom 𝑔) ⊆ (𝐴 𝐷)
108, 9eqsstri 4017 . . . . . . . 8 dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷)
11 sslin 4235 . . . . . . . 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 3992 . . . . . 6 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
14 disjdif 4472 . . . . . 6 ( 𝐷 ∩ (𝐴 𝐷)) = ∅
1513, 14sseqtri 4019 . . . . 5 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅
16 ss0 4399 . . . . 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 6595 . . . 4 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
1917, 18mpan2 688 . . 3 ((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
201, 2, 19syl2an 595 . 2 ((Fun 𝑓 ∧ Fun 𝑔) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
21 sbthlem.3 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2221funeqi 6570 . 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 395   = wceq 1540  wcel 2105  {cab 2708  Vcvv 3473  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323   cuni 4909  ccnv 5676  dom cdm 5677  cres 5679  cima 5680  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-fun 6546
This theorem is referenced by:  sbthlem9  9094
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