Theorem sbthlem1 9149

Description: Lemma for sbth 9159. (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 4963	. 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 2888	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
4		difss2 4161	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴)
5		ssconb 4165	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
6	5	exbiri 810	. . . . . . 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 406	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
10	3, 9	sylbi 217	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
11		elssuni 4961	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ ∪ 𝐷)
12		imass2 6132	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐷 → (𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷))
13		sscon 4166	. . . . 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 6132	. . . 4 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))
16		sscon 4166	. . . 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 4019	. 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	1, 18	mprgbir 3074	1 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∪ cuni 4931   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  sbthlem2  9150  sbthlem3  9151  sbthlem5  9153