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Theorem sbthlem8 9092
Description: Lemma for sbth 9095. (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
sbthlem8 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ 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 sbthlem8
StepHypRef Expression
1 funres11 6624 . . . 4 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funcnvcnv 6614 . . . . . 6 (Fun 𝑔 → Fun 𝑔)
3 funres11 6624 . . . . . 6 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
42, 3syl 17 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
54ad3antrrr 726 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → Fun (𝑔 ↾ (𝐴 𝐷)))
61, 5anim12i 611 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))))
7 df-ima 5688 . . . . . . . 8 (𝑓 𝐷) = ran (𝑓 𝐷)
8 df-rn 5686 . . . . . . . 8 ran (𝑓 𝐷) = dom (𝑓 𝐷)
97, 8eqtr2i 2759 . . . . . . 7 dom (𝑓 𝐷) = (𝑓 𝐷)
10 df-ima 5688 . . . . . . . . 9 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
11 df-rn 5686 . . . . . . . . 9 ran (𝑔 ↾ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
1210, 11eqtri 2758 . . . . . . . 8 (𝑔 “ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
13 sbthlem.1 . . . . . . . . 9 𝐴 ∈ V
14 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1513, 14sbthlem4 9088 . . . . . . . 8 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
1612, 15eqtr3id 2784 . . . . . . 7 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
17 ineq12 4206 . . . . . . 7 ((dom (𝑓 𝐷) = (𝑓 𝐷) ∧ dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷))) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
189, 16, 17sylancr 585 . . . . . 6 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
19 disjdif 4470 . . . . . 6 ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))) = ∅
2018, 19eqtrdi 2786 . . . . 5 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2120adantlll 714 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2221adantl 480 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
23 funun 6593 . . 3 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
246, 22, 23syl2anc 582 . 2 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
25 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2625cnveqi 5873 . . . 4 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
27 cnvun 6141 . . . 4 ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2826, 27eqtri 2758 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2928funeqi 6568 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
3024, 29sylibr 233 1 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321   cuni 4907  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  Fun wfun 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544
This theorem is referenced by:  sbthlem9  9093
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