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Theorem sbthlem2 8015
Description: Lemma for sbth 8024. (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 8014 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
4 imass2 5460 . . . . . . . 8 ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
5 sscon 3722 . . . . . . . 8 ((𝑓 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) → (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 𝐷)))
63, 4, 5mp2b 10 . . . . . . 7 (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 𝐷))
7 imass2 5460 . . . . . . 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 3722 . . . . . . 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 10 . . . . . 6 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
10 imassrn 5436 . . . . . . . 8 (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ ran 𝑔
11 sstr2 3590 . . . . . . . 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 3715 . . . . . . 7 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
14 ssconb 3721 . . . . . . 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 693 . . . . . 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 248 . . . . 5 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
1716, 13jctil 559 . . . 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 4763 . . . . 5 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ V
19 sseq1 3605 . . . . . 6 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑥𝐴 ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴))
20 imaeq2 5421 . . . . . . . . 9 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓𝑥) = (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2120difeq2d 3706 . . . . . . . 8 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐵 ∖ (𝑓𝑥)) = (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
2221imaeq2d 5425 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) = (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
23 difeq2 3700 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2422, 23sseq12d 3613 . . . . . 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 746 . . . . 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 3333 . . . 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 224 . . 3 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))})
2827, 2syl6eleqr 2709 . 2 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ 𝐷)
29 elssuni 4433 . 2 ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
3028, 29syl 17 1 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  cdif 3552  wss 3555   cuni 4402  ran crn 5075  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  sbthlem3  8016
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