Theorem sbthlem5 3353

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))}
sbthlem.3	⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))

Assertion

Ref	Expression
sbthlem5	⊢ ((dom f = A ∧ ran g ⊆ A) → dom H = A)

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

Proof of Theorem sbthlem5

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . . 9 ⊢ A ∈ V
2		sbthlem.2	. . . . . . . . 9 ⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
3	1, 2	sbthlem1 3349	. . . . . . . 8 ⊢ ∪D ⊆ (A ∖ (g “ (B ∖ (f “ ∪D))))
4		difss 1596	. . . . . . . 8 ⊢ (A ∖ (g “ (B ∖ (f “ ∪D)))) ⊆ A
5	3, 4	sstri 1512	. . . . . . 7 ⊢ ∪D ⊆ A
6		sseq2 1522	. . . . . . 7 ⊢ (dom f = A → (∪D ⊆ dom f ↔ ∪D ⊆ A))
7	5, 6	mpbiri 169	. . . . . 6 ⊢ (dom f = A → ∪D ⊆ dom f)
8		dfss 1493	. . . . . 6 ⊢ (∪D ⊆ dom f ↔ ∪D = (∪D ∩ dom f))
9	7, 8	sylib 173	. . . . 5 ⊢ (dom f = A → ∪D = (∪D ∩ dom f))
10	9	uneq1d 1610	. . . 4 ⊢ (dom f = A → (∪D ∪ (A ∖ ∪D)) = ((∪D ∩ dom f) ∪ (A ∖ ∪D)))
11		imassrn 2611	. . . . . . 7 ⊢ (g “ (B ∖ (f “ ∪D))) ⊆ ran g
12	1, 2	sbthlem3 3351	. . . . . . . 8 ⊢ (ran g ⊆ A → (g “ (B ∖ (f “ ∪D))) = (A ∖ ∪D))
13	12	sseq1d 1527	. . . . . . 7 ⊢ (ran g ⊆ A → ((g “ (B ∖ (f “ ∪D))) ⊆ ran g ↔ (A ∖ ∪D) ⊆ ran g))
14	11, 13	mpbii 168	. . . . . 6 ⊢ (ran g ⊆ A → (A ∖ ∪D) ⊆ ran g)
15		dfss 1493	. . . . . 6 ⊢ ((A ∖ ∪D) ⊆ ran g ↔ (A ∖ ∪D) = ((A ∖ ∪D) ∩ ran g))
16	14, 15	sylib 173	. . . . 5 ⊢ (ran g ⊆ A → (A ∖ ∪D) = ((A ∖ ∪D) ∩ ran g))
17	16	uneq2d 1611	. . . 4 ⊢ (ran g ⊆ A → ((∪D ∩ dom f) ∪ (A ∖ ∪D)) = ((∪D ∩ dom f) ∪ ((A ∖ ∪D) ∩ ran g)))
18	10, 17	sylan9eq 1144	. . 3 ⊢ ((dom f = A ∧ ran g ⊆ A) → (∪D ∪ (A ∖ ∪D)) = ((∪D ∩ dom f) ∪ ((A ∖ ∪D) ∩ ran g)))
19		sbthlem.3	. . . . 5 ⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
20	19	dmeqi 2532	. . . 4 ⊢ dom H = dom ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
21		dmun 2536	. . . 4 ⊢ dom ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))) = (dom (f ↾ ∪D) ∪ dom (◡g ↾ (A ∖ ∪D)))
22		dmres 2584	. . . . 5 ⊢ dom (f ↾ ∪D) = (∪D ∩ dom f)
23		dmres 2584	. . . . . 6 ⊢ dom (◡g ↾ (A ∖ ∪D)) = ((A ∖ ∪D) ∩ dom ◡g)
24		df-rn 2429	. . . . . . . 8 ⊢ ran g = dom ◡g
25	24	cleqcomi 1105	. . . . . . 7 ⊢ dom ◡g = ran g
26	25	ineq2i 1642	. . . . . 6 ⊢ ((A ∖ ∪D) ∩ dom ◡g) = ((A ∖ ∪D) ∩ ran g)
27	23, 26	eqtr 1119	. . . . 5 ⊢ dom (◡g ↾ (A ∖ ∪D)) = ((A ∖ ∪D) ∩ ran g)
28	22, 27	uneq12i 1609	. . . 4 ⊢ (dom (f ↾ ∪D) ∪ dom (◡g ↾ (A ∖ ∪D))) = ((∪D ∩ dom f) ∪ ((A ∖ ∪D) ∩ ran g))
29	20, 21, 28	3eqtr 1123	. . 3 ⊢ dom H = ((∪D ∩ dom f) ∪ ((A ∖ ∪D) ∩ ran g))
30	18, 29	syl6reqr 1143	. 2 ⊢ ((dom f = A ∧ ran g ⊆ A) → dom H = (∪D ∪ (A ∖ ∪D)))
31		ssundif 1764	. . 3 ⊢ (∪D ⊆ A ↔ (∪D ∪ (A ∖ ∪D)) = A)
32	5, 31	mpbi 164	. 2 ⊢ (∪D ∪ (A ∖ ∪D)) = A
33	30, 32	syl6eq 1140	1 ⊢ ((dom f = A ∧ ran g ⊆ A) → dom H = A)

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487  ∪cuni 1919  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413

This theorem is referenced by:  sbthlem9 3357

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

metamath.org