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Theorem sbthlem1 8111
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
sbthlem1 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 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 sbthlem1
StepHypRef Expression
1 unissb 4501 . 2 ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ↔ ∀𝑥𝐷 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
2 sbthlem.2 . . . . 5 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
32abeq2i 2764 . . . 4 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
4 difss2 3772 . . . . . . 7 ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴)
5 ssconb 3776 . . . . . . . 8 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
65exbiri 651 . . . . . . 7 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))))
74, 6syl5 34 . . . . . 6 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))))
87pm2.43d 53 . . . . 5 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))))
98imp 444 . . . 4 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
103, 9sylbi 207 . . 3 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
11 elssuni 4499 . . . . 5 (𝑥𝐷𝑥 𝐷)
12 imass2 5536 . . . . 5 (𝑥 𝐷 → (𝑓𝑥) ⊆ (𝑓 𝐷))
13 sscon 3777 . . . . 5 ((𝑓𝑥) ⊆ (𝑓 𝐷) → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
1411, 12, 133syl 18 . . . 4 (𝑥𝐷 → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
15 imass2 5536 . . . 4 ((𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))
16 sscon 3777 . . . 4 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
1714, 15, 163syl 18 . . 3 (𝑥𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
1810, 17sstrd 3646 . 2 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
191, 18mprgbir 2956 1 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604  wss 3607   cuni 4468  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
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-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  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:  sbthlem2  8112  sbthlem3  8113  sbthlem5  8115
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