Theorem sbthlem1 7199

Description: Lemma for isbth 7209. (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 3928	. 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 2342	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
4		difss2 3337	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴)
5		ssconb 3342	. . . . . . . 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 3926	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ ∪ 𝐷)
12		imass2 5119	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐷 → (𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷))
13		sscon 3343	. . . . 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 5119	. . . 4 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))
16		sscon 3343	. . . 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 3238	. 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	1, 18	mprgbir 2591	1 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {cab 2217  Vcvv 2803   ∖ cdif 3198   ⊆ wss 3201  ∪ cuni 3898   “ cima 4734

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  sbthlem2  7200  sbthlemi3  7201  sbthlemi5  7203