Theorem sbthlem3 3351

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
sbthlem3	⊢ (ran g ⊆ A → (g “ (B ∖ (f “ ∪D))) = (A ∖ ∪D))

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

Proof of Theorem sbthlem3

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . 6 ⊢ A ∈ V
2		sbthlem.2	. . . . . 6 ⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
3	1, 2	sbthlem2 3350	. . . . 5 ⊢ (ran g ⊆ A → (A ∖ (g “ (B ∖ (f “ ∪D)))) ⊆ ∪D)
4	1, 2	sbthlem1 3349	. . . . 5 ⊢ ∪D ⊆ (A ∖ (g “ (B ∖ (f “ ∪D))))
5	3, 4	jctil 240	. . . 4 ⊢ (ran g ⊆ A → (∪D ⊆ (A ∖ (g “ (B ∖ (f “ ∪D)))) ∧ (A ∖ (g “ (B ∖ (f “ ∪D)))) ⊆ ∪D))
6		eqss 1516	. . . 4 ⊢ (∪D = (A ∖ (g “ (B ∖ (f “ ∪D)))) ↔ (∪D ⊆ (A ∖ (g “ (B ∖ (f “ ∪D)))) ∧ (A ∖ (g “ (B ∖ (f “ ∪D)))) ⊆ ∪D))
7	5, 6	sylibr 175	. . 3 ⊢ (ran g ⊆ A → ∪D = (A ∖ (g “ (B ∖ (f “ ∪D)))))
8	7	difeq2d 1588	. 2 ⊢ (ran g ⊆ A → (A ∖ ∪D) = (A ∖ (A ∖ (g “ (B ∖ (f “ ∪D))))))
9		imassrn 2611	. . . 4 ⊢ (g “ (B ∖ (f “ ∪D))) ⊆ ran g
10		sstr2 1510	. . . 4 ⊢ ((g “ (B ∖ (f “ ∪D))) ⊆ ran g → (ran g ⊆ A → (g “ (B ∖ (f “ ∪D))) ⊆ A))
11	9, 10	ax-mp 6	. . 3 ⊢ (ran g ⊆ A → (g “ (B ∖ (f “ ∪D))) ⊆ A)
12		dfss4 1667	. . 3 ⊢ ((g “ (B ∖ (f “ ∪D))) ⊆ A ↔ (A ∖ (A ∖ (g “ (B ∖ (f “ ∪D))))) = (g “ (B ∖ (f “ ∪D))))
13	11, 12	sylib 173	. 2 ⊢ (ran g ⊆ A → (A ∖ (A ∖ (g “ (B ∖ (f “ ∪D))))) = (g “ (B ∖ (f “ ∪D))))
14	8, 13	eqtr2d 1129	1 ⊢ (ran g ⊆ A → (g “ (B ∖ (f “ ∪D))) = (A ∖ ∪D))

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  ∪cuni 1919  ran crn 2411   “ cima 2413

This theorem is referenced by:  sbthlem4 3352  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-rex 1206  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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