Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbthlem2 GIF version

Theorem sbthlem2 6863
 Description: Lemma for isbth 6872. (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
sbthlem2 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 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 sbthlem2
StepHypRef Expression
1 sbthlem.1 . . . . . . . . 9 𝐴 ∈ V
2 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
31, 2sbthlem1 6862 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
4 imass2 4927 . . . . . . . 8 ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
5 sscon 3217 . . . . . . . 8 ((𝑓 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) → (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 𝐷)))
63, 4, 5mp2b 8 . . . . . . 7 (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 𝐷))
7 imass2 4927 . . . . . . 7 ((𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 𝐷)) → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
8 sscon 3217 . . . . . . 7 ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))))
96, 7, 8mp2b 8 . . . . . 6 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
10 imassrn 4904 . . . . . . . 8 (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ ran 𝑔
11 sstr2 3111 . . . . . . . 8 ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ ran 𝑔 → (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ 𝐴))
1210, 11ax-mp 5 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ 𝐴)
13 difss 3209 . . . . . . 7 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
14 ssconb 3216 . . . . . . 7 (((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ 𝐴 ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴) → ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))))
1512, 13, 14sylancl 410 . . . . . 6 (ran 𝑔𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))))
169, 15mpbiri 167 . . . . 5 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
1716, 13jctil 310 . . . 4 (ran 𝑔𝐴 → ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
181, 13ssexi 4076 . . . . 5 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ V
19 sseq1 3127 . . . . . 6 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑥𝐴 ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴))
20 imaeq2 4889 . . . . . . . . 9 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓𝑥) = (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2120difeq2d 3201 . . . . . . . 8 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐵 ∖ (𝑓𝑥)) = (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
2221imaeq2d 4893 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) = (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
23 difeq2 3195 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2422, 23sseq12d 3135 . . . . . 6 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
2519, 24anbi12d 465 . . . . 5 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)) ↔ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
2618, 25elab 2834 . . . 4 ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))} ↔ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
2717, 26sylibr 133 . . 3 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))})
2827, 2eleqtrrdi 2235 . 2 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ 𝐷)
29 elssuni 3774 . 2 ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
3028, 29syl 14 1 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2112  {cab 2127  Vcvv 2691   ∖ cdif 3075   ⊆ wss 3078  ∪ cuni 3746  ran crn 4552   “ cima 4554 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-xp 4557  df-cnv 4559  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564 This theorem is referenced by:  sbthlemi3  6864
 Copyright terms: Public domain W3C validator