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Theorem sbthlem1 7955
 Description: Lemma for sbth 7965. (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 4405 . 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 2722 . . . 4 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
4 difss2 3701 . . . . . . 7 ((𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥) → (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴)
5 ssconb 3705 . . . . . . . 8 ((𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥)))
65exbiri 650 . . . . . . 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 51 . . . . 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 206 . . 3 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓𝑥)))))
11 elssuni 4403 . . . . 5 (𝑥𝐷𝑥 𝐷)
12 imass2 5420 . . . . 5 (𝑥 𝐷 → (𝑓𝑥) ⊆ (𝑓 𝐷))
13 sscon 3706 . . . . 5 ((𝑓𝑥) ⊆ (𝑓 𝐷) → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
1411, 12, 133syl 18 . . . 4 (𝑥𝐷 → (𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)))
15 imass2 5420 . . . 4 ((𝐵 ∖ (𝑓 𝐷)) ⊆ (𝐵 ∖ (𝑓𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓𝑥))))
16 sscon 3706 . . . 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 3578 . 2 (𝑥𝐷𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
191, 18mprgbir 2911 1 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  sbthlem2  7956  sbthlem3  7957  sbthlem5  7959
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