Theorem sbthlem8 3356

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
sbthlem8	⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → Fun ◡H)

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

Proof of Theorem sbthlem8

Step	Hyp	Ref	Expression
1		funun 2700	. . 3 ⊢ (((Fun ◡(f ↾ ∪D) ∧ Fun ◡(◡g ↾ (A ∖ ∪D))) ∧ (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ∅) → Fun (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D))))
2		funres11 2709	. . . 4 ⊢ (Fun ◡f → Fun ◡(f ↾ ∪D))
3		funcnvcnv 2701	. . . . . . 7 ⊢ (Fun g → Fun ◡◡g)
4		funres11 2709	. . . . . . 7 ⊢ (Fun ◡◡g → Fun ◡(◡g ↾ (A ∖ ∪D)))
5	3, 4	syl 12	. . . . . 6 ⊢ (Fun g → Fun ◡(◡g ↾ (A ∖ ∪D)))
6	5	adantr 306	. . . . 5 ⊢ ((Fun g ∧ dom g = B) → Fun ◡(◡g ↾ (A ∖ ∪D)))
7	6	ad2antll 320	. . . 4 ⊢ ((((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g) → Fun ◡(◡g ↾ (A ∖ ∪D)))
8	2, 7	anim12i 268	. . 3 ⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (Fun ◡(f ↾ ∪D) ∧ Fun ◡(◡g ↾ (A ∖ ∪D))))
9		sbthlem.1	. . . . . . . . . 10 ⊢ A ∈ V
10		sbthlem.2	. . . . . . . . . 10 ⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
11	9, 10	sbthlem4 3352	. . . . . . . . 9 ⊢ (((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g) → (◡g “ (A ∖ ∪D)) = (B ∖ (f “ ∪D)))
12		df-ima 2431	. . . . . . . . . 10 ⊢ (◡g “ (A ∖ ∪D)) = ran (◡g ↾ (A ∖ ∪D))
13		df-rn 2429	. . . . . . . . . 10 ⊢ ran (◡g ↾ (A ∖ ∪D)) = dom ◡(◡g ↾ (A ∖ ∪D))
14	12, 13	eqtr 1119	. . . . . . . . 9 ⊢ (◡g “ (A ∖ ∪D)) = dom ◡(◡g ↾ (A ∖ ∪D))
15	11, 14	syl5eqr 1138	. . . . . . . 8 ⊢ (((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g) → dom ◡(◡g ↾ (A ∖ ∪D)) = (B ∖ (f “ ∪D)))
16		df-ima 2431	. . . . . . . . 9 ⊢ (f “ ∪D) = ran (f ↾ ∪D)
17		df-rn 2429	. . . . . . . . 9 ⊢ ran (f ↾ ∪D) = dom ◡(f ↾ ∪D)
18	16, 17	eqtr2 1120	. . . . . . . 8 ⊢ dom ◡(f ↾ ∪D) = (f “ ∪D)
19	15, 18	jctil 240	. . . . . . 7 ⊢ (((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g) → (dom ◡(f ↾ ∪D) = (f “ ∪D) ∧ dom ◡(◡g ↾ (A ∖ ∪D)) = (B ∖ (f “ ∪D))))
20		ineq12 1640	. . . . . . 7 ⊢ ((dom ◡(f ↾ ∪D) = (f “ ∪D) ∧ dom ◡(◡g ↾ (A ∖ ∪D)) = (B ∖ (f “ ∪D))) → (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ((f “ ∪D) ∩ (B ∖ (f “ ∪D))))
21	19, 20	syl 12	. . . . . 6 ⊢ (((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g) → (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ((f “ ∪D) ∩ (B ∖ (f “ ∪D))))
22		difdisj 1758	. . . . . 6 ⊢ ((f “ ∪D) ∩ (B ∖ (f “ ∪D))) = ∅
23	21, 22	syl6eq 1140	. . . . 5 ⊢ (((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g) → (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ∅)
24	23	adantlll 313	. . . 4 ⊢ ((((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g) → (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ∅)
25	24	adantl 305	. . 3 ⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (dom ◡(f ↾ ∪D) ∩ dom ◡(◡g ↾ (A ∖ ∪D))) = ∅)
26	1, 8, 25	sylanc 361	. 2 ⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → Fun (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D))))
27		sbthlem.3	. . . . 5 ⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
28		cnveq 2513	. . . . 5 ⊢ (H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))) → ◡H = ◡((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))))
29	27, 28	ax-mp 6	. . . 4 ⊢ ◡H = ◡((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
30		cnvun 2642	. . . 4 ⊢ ◡((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D))) = (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D)))
31	29, 30	eqtr 1119	. . 3 ⊢ ◡H = (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D)))
32		funeq 2683	. . 3 ⊢ (◡H = (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D))) → (Fun ◡H ↔ Fun (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D)))))
33	31, 32	ax-mp 6	. 2 ⊢ (Fun ◡H ↔ Fun (◡(f ↾ ∪D) ∪ ◡(◡g ↾ (A ∖ ∪D))))
34	26, 33	sylibr 175	1 ⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ 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  ran crn 2411   ↾ 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-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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

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