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Theorem sbthlem7 8668
 Description: Lemma for sbth 8672. (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 6382 . . 3 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funres 6382 . . 3 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
3 dmres 5850 . . . . . . . . 9 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
4 inss1 4135 . . . . . . . . 9 ( 𝐷 ∩ dom 𝑓) ⊆ 𝐷
53, 4eqsstri 3928 . . . . . . . 8 dom (𝑓 𝐷) ⊆ 𝐷
6 ssrin 4140 . . . . . . . 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 5850 . . . . . . . . 9 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
9 inss1 4135 . . . . . . . . 9 ((𝐴 𝐷) ∩ dom 𝑔) ⊆ (𝐴 𝐷)
108, 9eqsstri 3928 . . . . . . . 8 dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷)
11 sslin 4141 . . . . . . . 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 3903 . . . . . 6 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
14 disjdif 4371 . . . . . 6 ( 𝐷 ∩ (𝐴 𝐷)) = ∅
1513, 14sseqtri 3930 . . . . 5 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅
16 ss0 4297 . . . . 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 6386 . . . 4 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
1917, 18mpan2 690 . . 3 ((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
201, 2, 19syl2an 598 . 2 ((Fun 𝑓 ∧ Fun 𝑔) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
21 sbthlem.3 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2221funeqi 6361 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
2320, 22sylibr 237 1 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ∪ cuni 4801  ◡ccnv 5527  dom cdm 5528   ↾ cres 5530   “ cima 5531  Fun wfun 6334 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-fun 6342 This theorem is referenced by:  sbthlem9  8670
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