Theorem sbthlem7 3355

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
sbthlem7	⊢ ((Fun f ∧ Fun ◡g) → Fun H)

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

Proof of Theorem sbthlem7

Step	Hyp	Ref	Expression
1		dmres 2584	. . . . . . . . 9 ⊢ dom (f ↾ ∪D) = (∪D ∩ dom f)
2		inss1 1657	. . . . . . . . 9 ⊢ (∪D ∩ dom f) ⊆ ∪D
3	1, 2	eqsstr 1530	. . . . . . . 8 ⊢ dom (f ↾ ∪D) ⊆ ∪D
4		ssrin 1661	. . . . . . . 8 ⊢ (dom (f ↾ ∪D) ⊆ ∪D → (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ (∪D ∩ dom (◡g ↾ (A ∖ ∪D))))
5	3, 4	ax-mp 6	. . . . . . 7 ⊢ (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ (∪D ∩ dom (◡g ↾ (A ∖ ∪D)))
6		dmres 2584	. . . . . . . . 9 ⊢ dom (◡g ↾ (A ∖ ∪D)) = ((A ∖ ∪D) ∩ dom ◡g)
7		inss1 1657	. . . . . . . . 9 ⊢ ((A ∖ ∪D) ∩ dom ◡g) ⊆ (A ∖ ∪D)
8	6, 7	eqsstr 1530	. . . . . . . 8 ⊢ dom (◡g ↾ (A ∖ ∪D)) ⊆ (A ∖ ∪D)
9		sslin 1662	. . . . . . . 8 ⊢ (dom (◡g ↾ (A ∖ ∪D)) ⊆ (A ∖ ∪D) → (∪D ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ (∪D ∩ (A ∖ ∪D)))
10	8, 9	ax-mp 6	. . . . . . 7 ⊢ (∪D ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ (∪D ∩ (A ∖ ∪D))
11	5, 10	sstri 1512	. . . . . 6 ⊢ (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ (∪D ∩ (A ∖ ∪D))
12		difdisj 1758	. . . . . 6 ⊢ (∪D ∩ (A ∖ ∪D)) = ∅
13	11, 12	sseqtr 1532	. . . . 5 ⊢ (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ ∅
14		ss0 1727	. . . . 5 ⊢ ((dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) ⊆ ∅ → (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) = ∅)
15	13, 14	ax-mp 6	. . . 4 ⊢ (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) = ∅
16		funun 2700	. . . 4 ⊢ (((Fun (f ↾ ∪D) ∧ Fun (◡g ↾ (A ∖ ∪D))) ∧ (dom (f ↾ ∪D) ∩ dom (◡g ↾ (A ∖ ∪D))) = ∅) → Fun ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))))
17	15, 16	mpan2 519	. . 3 ⊢ ((Fun (f ↾ ∪D) ∧ Fun (◡g ↾ (A ∖ ∪D))) → Fun ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))))
18		funres 2697	. . 3 ⊢ (Fun f → Fun (f ↾ ∪D))
19		funres 2697	. . 3 ⊢ (Fun ◡g → Fun (◡g ↾ (A ∖ ∪D)))
20	17, 18, 19	syl2an 349	. 2 ⊢ ((Fun f ∧ Fun ◡g) → Fun ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))))
21		sbthlem.3	. . 3 ⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
22		funeq 2683	. . 3 ⊢ (H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))) → (Fun H ↔ Fun ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))))
23	21, 22	ax-mp 6	. 2 ⊢ (Fun H ↔ Fun ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))))
24	20, 23	sylibr 175	1 ⊢ ((Fun f ∧ Fun ◡g) → Fun H)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919  ^◡ccnv 2409  dom cdm 2410   ↾ cres 2412   “ cima 2413  Fun wfun 2416

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-eu 1009  df-mo 1010  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-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432

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