Theorem sbthlem1 7120

Description: Lemma for isbth 7130. (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 3917	. 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	abeq2i 2340	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
4		difss2 3332	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴)
5		ssconb 3337	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
6	5	exbiri 382	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))))
7	4, 6	syl5 32	. . . . . 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 50	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))))
9	8	imp 124	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
10	3, 9	sylbi 121	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
11		elssuni 3915	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ ∪ 𝐷)
12		imass2 5103	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐷 → (𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷))
13		sscon 3338	. . . . 5 ⊢ ((𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
14	11, 12, 13	3syl 17	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
15		imass2 5103	. . . 4 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))
16		sscon 3338	. . . 4 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
17	14, 15, 16	3syl 17	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
18	10, 17	sstrd 3234	. 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	1, 18	mprgbir 2588	1 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  Vcvv 2799   ∖ cdif 3194   ⊆ wss 3197  ∪ cuni 3887   “ cima 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by:  sbthlem2  7121  sbthlemi3  7122  sbthlemi5  7124