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Theorem sbthlem8 8319
Description: Lemma for sbth 8322. (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 6177 . . . 4 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funcnvcnv 6167 . . . . . 6 (Fun 𝑔 → Fun 𝑔)
3 funres11 6177 . . . . . 6 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
42, 3syl 17 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
54ad3antrrr 722 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → Fun (𝑔 ↾ (𝐴 𝐷)))
61, 5anim12i 607 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))))
7 df-ima 5325 . . . . . . . 8 (𝑓 𝐷) = ran (𝑓 𝐷)
8 df-rn 5323 . . . . . . . 8 ran (𝑓 𝐷) = dom (𝑓 𝐷)
97, 8eqtr2i 2822 . . . . . . 7 dom (𝑓 𝐷) = (𝑓 𝐷)
10 df-ima 5325 . . . . . . . . 9 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
11 df-rn 5323 . . . . . . . . 9 ran (𝑔 ↾ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
1210, 11eqtri 2821 . . . . . . . 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 8315 . . . . . . . 8 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
1612, 15syl5eqr 2847 . . . . . . 7 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
17 ineq12 4007 . . . . . . 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 582 . . . . . 6 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
19 disjdif 4234 . . . . . 6 ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))) = ∅
2018, 19syl6eq 2849 . . . . 5 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2120adantlll 710 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2221adantl 474 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
23 funun 6146 . . 3 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
246, 22, 23syl2anc 580 . 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 5500 . . . 4 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
27 cnvun 5755 . . . 4 ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2826, 27eqtri 2821 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2928funeqi 6122 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
3024, 29sylibr 226 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 385   = wceq 1653  wcel 2157  {cab 2785  Vcvv 3385  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4115   cuni 4628  ccnv 5311  dom cdm 5312  ran crn 5313  cres 5314  cima 5315  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-fun 6103
This theorem is referenced by:  sbthlem9  8320
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