Theorem sbthlem1 3349

Description: Lemma for Schroeder-Bernstein Theorem.

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 group(s):   x,A   x,B   x,D   x,f   x,g

Proof of Theorem sbthlem1

Step	Hyp	Ref	Expression
1		unissb 1941	. 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	cleqabi 1176	. . . 4 ⊢ (x ∈ D ↔ (x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x)))
4		ssconb 1598	. . . . . . . . 9 ⊢ ((x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ A) → (x ⊆ (A ∖ (g “ (B ∖ (f “ x)))) ↔ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x)))
5	4	biimprd 136	. . . . . . . 8 ⊢ ((x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ A) → ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → x ⊆ (A ∖ (g “ (B ∖ (f “ x))))))
6	5	exp 291	. . . . . . 7 ⊢ (x ⊆ A → ((g “ (B ∖ (f “ x))) ⊆ A → ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → x ⊆ (A ∖ (g “ (B ∖ (f “ x)))))))
7		difss 1596	. . . . . . . 8 ⊢ (A ∖ x) ⊆ A
8		sstr2 1510	. . . . . . . 8 ⊢ ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → ((A ∖ x) ⊆ A → (g “ (B ∖ (f “ x))) ⊆ A))
9	7, 8	mpi 44	. . . . . . 7 ⊢ ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → (g “ (B ∖ (f “ x))) ⊆ A)
10	6, 9	syl5 22	. . . . . 6 ⊢ (x ⊆ A → ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → x ⊆ (A ∖ (g “ (B ∖ (f “ x)))))))
11	10	pm2.43d 59	. . . . 5 ⊢ (x ⊆ A → ((g “ (B ∖ (f “ x))) ⊆ (A ∖ x) → x ⊆ (A ∖ (g “ (B ∖ (f “ x))))))
12	11	imp 277	. . . 4 ⊢ ((x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x)) → x ⊆ (A ∖ (g “ (B ∖ (f “ x)))))
13	3, 12	sylbi 174	. . 3 ⊢ (x ∈ D → x ⊆ (A ∖ (g “ (B ∖ (f “ x)))))
14		elssuni 1940	. . . . 5 ⊢ (x ∈ D → x ⊆ ∪D)
15		imass2 2622	. . . . 5 ⊢ (x ⊆ ∪D → (f “ x) ⊆ (f “ ∪D))
16		sscon 1599	. . . . 5 ⊢ ((f “ x) ⊆ (f “ ∪D) → (B ∖ (f “ ∪D)) ⊆ (B ∖ (f “ x)))
17	14, 15, 16	3syl 21	. . . 4 ⊢ (x ∈ D → (B ∖ (f “ ∪D)) ⊆ (B ∖ (f “ x)))
18		imass2 2622	. . . 4 ⊢ ((B ∖ (f “ ∪D)) ⊆ (B ∖ (f “ x)) → (g “ (B ∖ (f “ ∪D))) ⊆ (g “ (B ∖ (f “ x))))
19		sscon 1599	. . . 4 ⊢ ((g “ (B ∖ (f “ ∪D))) ⊆ (g “ (B ∖ (f “ x))) → (A ∖ (g “ (B ∖ (f “ x)))) ⊆ (A ∖ (g “ (B ∖ (f “ ∪D)))))
20	17, 18, 19	3syl 21	. . 3 ⊢ (x ∈ D → (A ∖ (g “ (B ∖ (f “ x)))) ⊆ (A ∖ (g “ (B ∖ (f “ ∪D)))))
21	13, 20	sstrd 1513	. 2 ⊢ (x ∈ D → x ⊆ (A ∖ (g “ (B ∖ (f “ ∪D)))))
22	1, 21	mprgbir 1250	1 ⊢ ∪D ⊆ (A ∖ (g “ (B ∖ (f “ ∪D))))

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  ∪cuni 1919   “ cima 2413

This theorem is referenced by:  sbthlem2 3350  sbthlem3 3351  sbthlem5 3353

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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