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Theorem sbthlem1 7831
Description: Lemma for sbth 7841. (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 4303 . 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 2626 . . . 4 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
4 difss2 3605 . . . . . . 7 ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴)
5 ssconb 3609 . . . . . . . 8 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
65exbiri 649 . . . . . . 7 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))))
74, 6syl5 33 . . . . . 6 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))))
87pm2.43d 50 . . . . 5 (𝑥𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))))
98imp 443 . . . 4 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
103, 9sylbi 205 . . 3 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
11 elssuni 4301 . . . . 5 (𝑥𝐷𝑥 𝐷)
12 imass2 5311 . . . . 5 (𝑥 𝐷 → (𝑓𝑥) ⊆ (𝑓 𝐷))
13 sscon 3610 . . . . 5 ((𝑓𝑥) ⊆ (𝑓 𝐷) → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
1411, 12, 133syl 18 . . . 4 (𝑥𝐷 → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
15 imass2 5311 . . . 4 ((𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))
16 sscon 3610 . . . 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 3482 . 2 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
191, 18mprgbir 2815 1 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  {cab 2500  Vcvv 3077  cdif 3441  wss 3444   cuni 4270  cima 4935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-xp 4938  df-cnv 4940  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945
This theorem is referenced by:  sbthlem2  7832  sbthlem3  7833  sbthlem5  7835
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