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Theorem sbthlemi3 6668
Description: Lemma for isbth 6676. (Contributed by NM, 22-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
Assertion
Ref Expression
sbthlemi3 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)

Proof of Theorem sbthlemi3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbthlem.1 . . . . . . 7 𝐴 ∈ V
2 sbthlem.2 . . . . . . 7 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
31, 2sbthlem2 6667 . . . . . 6 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
41, 2sbthlem1 6666 . . . . . 6 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
53, 4jctil 305 . . . . 5 (ran 𝑔𝐴 → ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
6 eqss 3040 . . . . 5 ( 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ↔ ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
75, 6sylibr 132 . . . 4 (ran 𝑔𝐴 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
87difeq2d 3118 . . 3 (ran 𝑔𝐴 → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
98adantl 271 . 2 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
10 imassrn 4785 . . . . 5 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
11 sstr2 3032 . . . . 5 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔 → (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴))
1210, 11ax-mp 7 . . . 4 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴)
13 exmidexmid 4031 . . . . . . 7 (EXMIDDECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
14 dcimpstab 790 . . . . . . 7 (DECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) → STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1513, 14syl 14 . . . . . 6 (EXMIDSTAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1615alrimiv 1802 . . . . 5 (EXMID → ∀𝑦STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
17 dfss4st 3232 . . . . 5 (∀𝑦STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) → ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
1816, 17syl 14 . . . 4 (EXMID → ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
1912, 18syl5ib 152 . . 3 (EXMID → (ran 𝑔𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
2019imp 122 . 2 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
219, 20eqtr2d 2121 1 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  STAB wstab 775  DECID wdc 780  wal 1287   = wceq 1289  wcel 1438  {cab 2074  Vcvv 2619  cdif 2996  wss 2999   cuni 3653  EXMIDwem 4029  ran crn 4439  cima 4441
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-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by:  sbthlemi4  6669  sbthlemi5  6670
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