Theorem sbthlem1 9085

Description: Lemma for sbth 9095. (Contributed by NM, 22-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}

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

Step	Hyp	Ref	Expression
1		unissb 4942	. 2 ⊢ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ↔ ∀𝑥 ∈ 𝐷 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
2		sbthlem.2	. . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3	2	eqabri 2875	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
4		difss2 4132	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴)
5		ssconb 4136	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
6	5	exbiri 807	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))))
7	4, 6	syl5 34	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))))
8	7	pm2.43d 53	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))))
9	8	imp 405	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
10	3, 9	sylbi 216	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
11		elssuni 4940	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ ∪ 𝐷)
12		imass2 6100	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐷 → (𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷))
13		sscon 4137	. . . . 5 ⊢ ((𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
14	11, 12, 13	3syl 18	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
15		imass2 6100	. . . 4 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))
16		sscon 4137	. . . 4 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
17	14, 15, 16	3syl 18	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
18	10, 17	sstrd 3991	. 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	1, 18	mprgbir 3066	1 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  Vcvv 3472   ∖ cdif 3944   ⊆ wss 3947  ∪ cuni 4907   “ cima 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by:  sbthlem2  9086  sbthlem3  9087  sbthlem5  9089