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Theorem sbthlemi6 6671
Description: Lemma for isbth 6676. (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
sbthlemi6 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
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 sbthlemi6
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpll 496 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → EXMID)
2 simprll 504 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝑔 = 𝐵)
3 simprlr 505 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝑔𝐴)
4 simprr 499 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝑔)
5 df-ima 4451 . . . . . 6 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
6 sbthlem.1 . . . . . . 7 𝐴 ∈ V
7 sbthlem.2 . . . . . . 7 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
86, 7sbthlemi4 6669 . . . . . 6 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
95, 8syl5reqr 2135 . . . . 5 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
109uneq2d 3154 . . . 4 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))))
11 rnun 4840 . . . . 5 ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
12 sbthlem.3 . . . . . 6 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
1312rneqi 4663 . . . . 5 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
14 df-ima 4451 . . . . . 6 (𝑓 𝐷) = ran (𝑓 𝐷)
1514uneq1i 3150 . . . . 5 ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
1611, 13, 153eqtr4i 2118 . . . 4 ran 𝐻 = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
1710, 16syl6reqr 2139 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
181, 2, 3, 4, 17syl121anc 1179 . 2 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
19 imassrn 4785 . . . . . . 7 (𝑓 𝐷) ⊆ ran 𝑓
20 sstr2 3032 . . . . . . 7 ((𝑓 𝐷) ⊆ ran 𝑓 → (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵))
2119, 20ax-mp 7 . . . . . 6 (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵)
2221adantl 271 . . . . 5 ((EXMID ∧ ran 𝑓𝐵) → (𝑓 𝐷) ⊆ 𝐵)
23 exmidexmid 4031 . . . . . . . . 9 (EXMIDDECID 𝑦 ∈ (𝑓 𝐷))
2423ralrimivw 2447 . . . . . . . 8 (EXMID → ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷))
2524biantrud 298 . . . . . . 7 (EXMID → ((𝑓 𝐷) ⊆ 𝐵 ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷))))
26 undifdcss 6633 . . . . . . 7 (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷)))
2725, 26syl6rbbr 197 . . . . . 6 (EXMID → (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ (𝑓 𝐷) ⊆ 𝐵))
2827adantr 270 . . . . 5 ((EXMID ∧ ran 𝑓𝐵) → (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ (𝑓 𝐷) ⊆ 𝐵))
2922, 28mpbird 165 . . . 4 ((EXMID ∧ ran 𝑓𝐵) → 𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
3029eqcomd 2093 . . 3 ((EXMID ∧ ran 𝑓𝐵) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
3130adantr 270 . 2 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
3218, 31eqtrd 2120 1 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  DECID wdc 780  w3a 924   = wceq 1289  wcel 1438  {cab 2074  wral 2359  Vcvv 2619  cdif 2996  cun 2997  wss 2999   cuni 3653  EXMIDwem 4029  ccnv 4437  dom cdm 4438  ran crn 4439  cres 4440  cima 4441  Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  sbthlemi9  6674
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