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Theorem sbthlem2 8112
Description: Lemma for sbth 8121. (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 8111 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
4 imass2 5536 . . . . . . . 8 ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
5 sscon 3777 . . . . . . . 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 5536 . . . . . . 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 3777 . . . . . . 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 5512 . . . . . . . 8 (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))) ⊆ ran 𝑔
11 sstr2 3643 . . . . . . . 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 3770 . . . . . . 7 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
14 ssconb 3776 . . . . . . 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 695 . . . . . 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 4836 . . . . 5 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ V
19 sseq1 3659 . . . . . 6 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑥𝐴 ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴))
20 imaeq2 5497 . . . . . . . . 9 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑓𝑥) = (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2120difeq2d 3761 . . . . . . . 8 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐵 ∖ (𝑓𝑥)) = (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))))
2221imaeq2d 5501 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) = (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))))
23 difeq2 3755 . . . . . . 7 (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
2422, 23sseq12d 3667 . . . . . 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 747 . . . . 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 3382 . . . 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 2741 . 2 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∈ 𝐷)
29 elssuni 4499 . 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 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604  wss 3607   cuni 4468  ran crn 5144  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  sbthlem3  8113
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