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Theorem sbthlemi3 6813
Description: Lemma for isbth 6821. (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 6812 . . . . . 6 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
41, 2sbthlem1 6811 . . . . . 6 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
53, 4jctil 308 . . . . 5 (ran 𝑔𝐴 → ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
6 eqss 3080 . . . . 5 ( 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ↔ ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
75, 6sylibr 133 . . . 4 (ran 𝑔𝐴 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
87difeq2d 3162 . . 3 (ran 𝑔𝐴 → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
98adantl 273 . 2 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
10 imassrn 4860 . . . . 5 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
11 sstr2 3072 . . . . 5 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔 → (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴))
1210, 11ax-mp 5 . . . 4 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴)
13 exmidexmid 4088 . . . . . . 7 (EXMIDDECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
14 dcstab 812 . . . . . . 7 (DECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) → STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1513, 14syl 14 . . . . . 6 (EXMIDSTAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1615alrimiv 1828 . . . . 5 (EXMID → ∀𝑦STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
17 dfss4st 3277 . . . . 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 153 . . 3 (EXMID → (ran 𝑔𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
2019imp 123 . 2 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
219, 20eqtr2d 2149 1 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  STAB wstab 798  DECID wdc 802  wal 1312   = wceq 1314  wcel 1463  {cab 2101  Vcvv 2658  cdif 3036  wss 3039   cuni 3704  EXMIDwem 4086  ran crn 4508  cima 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-exmid 4087  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  sbthlemi4  6814  sbthlemi5  6815
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