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Theorem sbthlem8 8118
Description: Lemma for sbth 8121. (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 6004 . . . 4 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funcnvcnv 5994 . . . . . 6 (Fun 𝑔 → Fun 𝑔)
3 funres11 6004 . . . . . 6 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
42, 3syl 17 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
54ad3antrrr 766 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → Fun (𝑔 ↾ (𝐴 𝐷)))
61, 5anim12i 589 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))))
7 df-ima 5156 . . . . . . . 8 (𝑓 𝐷) = ran (𝑓 𝐷)
8 df-rn 5154 . . . . . . . 8 ran (𝑓 𝐷) = dom (𝑓 𝐷)
97, 8eqtr2i 2674 . . . . . . 7 dom (𝑓 𝐷) = (𝑓 𝐷)
10 df-ima 5156 . . . . . . . . 9 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
11 df-rn 5154 . . . . . . . . 9 ran (𝑔 ↾ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
1210, 11eqtri 2673 . . . . . . . 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 8114 . . . . . . . 8 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
1612, 15syl5eqr 2699 . . . . . . 7 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
17 ineq12 3842 . . . . . . 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 696 . . . . . 6 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
19 disjdif 4073 . . . . . 6 ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))) = ∅
2018, 19syl6eq 2701 . . . . 5 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2120adantlll 754 . . . 4 ((((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
2221adantl 481 . . 3 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
23 funun 5970 . . 3 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
246, 22, 23syl2anc 694 . 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 5329 . . . 4 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
27 cnvun 5573 . . . 4 ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2826, 27eqtri 2673 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2928funeqi 5947 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
3024, 29sylibr 224 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 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948   cuni 4468  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928
This theorem is referenced by:  sbthlem9  8119
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