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Theorem sbthlem1 9034
Description: Lemma for sbth 9044. (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 4905 . 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))}
32eqabi 2882 . . . 4 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
4 difss2 4098 . . . . . . 7 ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴)
5 ssconb 4102 . . . . . . . 8 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
65exbiri 810 . . . . . . 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 408 . . . 4 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
103, 9sylbi 216 . . 3 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
11 elssuni 4903 . . . . 5 (𝑥𝐷𝑥 𝐷)
12 imass2 6059 . . . . 5 (𝑥 𝐷 → (𝑓𝑥) ⊆ (𝑓 𝐷))
13 sscon 4103 . . . . 5 ((𝑓𝑥) ⊆ (𝑓 𝐷) → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
1411, 12, 133syl 18 . . . 4 (𝑥𝐷 → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
15 imass2 6059 . . . 4 ((𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))
16 sscon 4103 . . . 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 3959 . 2 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
191, 18mprgbir 3072 1 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2714  Vcvv 3448  cdif 3912  wss 3915   cuni 4870  cima 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  sbthlem2  9035  sbthlem3  9036  sbthlem5  9038
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