Theorem imadif 5172

Description: The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)

Assertion

Ref	Expression
imadif	⊢ (Fun ◡F → (F “ (A ∖ B)) = ((F “ A) ∖ (F “ B)))

Proof of Theorem imadif

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anandir 802	. . . . . . 7 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) ↔ ((x ∈ A ∧ xFy) ∧ (¬ x ∈ B ∧ xFy)))
2	1	exbii 1582	. . . . . 6 ⊢ (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) ↔ ∃x((x ∈ A ∧ xFy) ∧ (¬ x ∈ B ∧ xFy)))
3		19.40 1609	. . . . . 6 ⊢ (∃x((x ∈ A ∧ xFy) ∧ (¬ x ∈ B ∧ xFy)) → (∃x(x ∈ A ∧ xFy) ∧ ∃x(¬ x ∈ B ∧ xFy)))
4	2, 3	sylbi 187	. . . . 5 ⊢ (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) → (∃x(x ∈ A ∧ xFy) ∧ ∃x(¬ x ∈ B ∧ xFy)))
5		nfv 1619	. . . . . . . . . 10 ⊢ ℲxFun ◡F
6		nfe1 1732	. . . . . . . . . 10 ⊢ Ⅎx∃x(xFy ∧ ¬ x ∈ B)
7	5, 6	nfan 1824	. . . . . . . . 9 ⊢ Ⅎx(Fun ◡F ∧ ∃x(xFy ∧ ¬ x ∈ B))
8		funmo 5126	. . . . . . . . . . . . 13 ⊢ (Fun ◡F → ∃*x y◡Fx)
9		brcnv 4893	. . . . . . . . . . . . . 14 ⊢ (y◡Fx ↔ xFy)
10	9	mobii 2240	. . . . . . . . . . . . 13 ⊢ (∃*x y◡Fx ↔ ∃*x xFy)
11	8, 10	sylib 188	. . . . . . . . . . . 12 ⊢ (Fun ◡F → ∃*x xFy)
12		mopick 2266	. . . . . . . . . . . 12 ⊢ ((∃*x xFy ∧ ∃x(xFy ∧ ¬ x ∈ B)) → (xFy → ¬ x ∈ B))
13	11, 12	sylan 457	. . . . . . . . . . 11 ⊢ ((Fun ◡F ∧ ∃x(xFy ∧ ¬ x ∈ B)) → (xFy → ¬ x ∈ B))
14	13	con2d 107	. . . . . . . . . 10 ⊢ ((Fun ◡F ∧ ∃x(xFy ∧ ¬ x ∈ B)) → (x ∈ B → ¬ xFy))
15		imnan 411	. . . . . . . . . 10 ⊢ ((x ∈ B → ¬ xFy) ↔ ¬ (x ∈ B ∧ xFy))
16	14, 15	sylib 188	. . . . . . . . 9 ⊢ ((Fun ◡F ∧ ∃x(xFy ∧ ¬ x ∈ B)) → ¬ (x ∈ B ∧ xFy))
17	7, 16	alrimi 1765	. . . . . . . 8 ⊢ ((Fun ◡F ∧ ∃x(xFy ∧ ¬ x ∈ B)) → ∀x ¬ (x ∈ B ∧ xFy))
18	17	ex 423	. . . . . . 7 ⊢ (Fun ◡F → (∃x(xFy ∧ ¬ x ∈ B) → ∀x ¬ (x ∈ B ∧ xFy)))
19		exancom 1586	. . . . . . 7 ⊢ (∃x(xFy ∧ ¬ x ∈ B) ↔ ∃x(¬ x ∈ B ∧ xFy))
20		alnex 1543	. . . . . . 7 ⊢ (∀x ¬ (x ∈ B ∧ xFy) ↔ ¬ ∃x(x ∈ B ∧ xFy))
21	18, 19, 20	3imtr3g 260	. . . . . 6 ⊢ (Fun ◡F → (∃x(¬ x ∈ B ∧ xFy) → ¬ ∃x(x ∈ B ∧ xFy)))
22	21	anim2d 548	. . . . 5 ⊢ (Fun ◡F → ((∃x(x ∈ A ∧ xFy) ∧ ∃x(¬ x ∈ B ∧ xFy)) → (∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy))))
23	4, 22	syl5 28	. . . 4 ⊢ (Fun ◡F → (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) → (∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy))))
24		19.29r 1597	. . . . . 6 ⊢ ((∃x(x ∈ A ∧ xFy) ∧ ∀x ¬ (x ∈ B ∧ xFy)) → ∃x((x ∈ A ∧ xFy) ∧ ¬ (x ∈ B ∧ xFy)))
25	20, 24	sylan2br 462	. . . . 5 ⊢ ((∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy)) → ∃x((x ∈ A ∧ xFy) ∧ ¬ (x ∈ B ∧ xFy)))
26		andi 837	. . . . . . 7 ⊢ (((x ∈ A ∧ xFy) ∧ (¬ x ∈ B ∨ ¬ xFy)) ↔ (((x ∈ A ∧ xFy) ∧ ¬ x ∈ B) ∨ ((x ∈ A ∧ xFy) ∧ ¬ xFy)))
27		ianor 474	. . . . . . . 8 ⊢ (¬ (x ∈ B ∧ xFy) ↔ (¬ x ∈ B ∨ ¬ xFy))
28	27	anbi2i 675	. . . . . . 7 ⊢ (((x ∈ A ∧ xFy) ∧ ¬ (x ∈ B ∧ xFy)) ↔ ((x ∈ A ∧ xFy) ∧ (¬ x ∈ B ∨ ¬ xFy)))
29		an32 773	. . . . . . . 8 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) ↔ ((x ∈ A ∧ xFy) ∧ ¬ x ∈ B))
30		pm3.24 852	. . . . . . . . . . 11 ⊢ ¬ (xFy ∧ ¬ xFy)
31	30	intnan 880	. . . . . . . . . 10 ⊢ ¬ (x ∈ A ∧ (xFy ∧ ¬ xFy))
32		anass 630	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ xFy) ∧ ¬ xFy) ↔ (x ∈ A ∧ (xFy ∧ ¬ xFy)))
33	31, 32	mtbir 290	. . . . . . . . 9 ⊢ ¬ ((x ∈ A ∧ xFy) ∧ ¬ xFy)
34	33	biorfi 396	. . . . . . . 8 ⊢ (((x ∈ A ∧ xFy) ∧ ¬ x ∈ B) ↔ (((x ∈ A ∧ xFy) ∧ ¬ x ∈ B) ∨ ((x ∈ A ∧ xFy) ∧ ¬ xFy)))
35	29, 34	bitri 240	. . . . . . 7 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) ↔ (((x ∈ A ∧ xFy) ∧ ¬ x ∈ B) ∨ ((x ∈ A ∧ xFy) ∧ ¬ xFy)))
36	26, 28, 35	3bitr4i 268	. . . . . 6 ⊢ (((x ∈ A ∧ xFy) ∧ ¬ (x ∈ B ∧ xFy)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
37	36	exbii 1582	. . . . 5 ⊢ (∃x((x ∈ A ∧ xFy) ∧ ¬ (x ∈ B ∧ xFy)) ↔ ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
38	25, 37	sylib 188	. . . 4 ⊢ ((∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy)) → ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
39	23, 38	impbid1 194	. . 3 ⊢ (Fun ◡F → (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy) ↔ (∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy))))
40		elima2 4756	. . . 4 ⊢ (y ∈ (F “ (A ∖ B)) ↔ ∃x(x ∈ (A ∖ B) ∧ xFy))
41		eldif 3222	. . . . . 6 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
42	41	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∖ B) ∧ xFy) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
43	42	exbii 1582	. . . 4 ⊢ (∃x(x ∈ (A ∖ B) ∧ xFy) ↔ ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
44	40, 43	bitri 240	. . 3 ⊢ (y ∈ (F “ (A ∖ B)) ↔ ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ xFy))
45		eldif 3222	. . . 4 ⊢ (y ∈ ((F “ A) ∖ (F “ B)) ↔ (y ∈ (F “ A) ∧ ¬ y ∈ (F “ B)))
46		elima2 4756	. . . . 5 ⊢ (y ∈ (F “ A) ↔ ∃x(x ∈ A ∧ xFy))
47		elima2 4756	. . . . . 6 ⊢ (y ∈ (F “ B) ↔ ∃x(x ∈ B ∧ xFy))
48	47	notbii 287	. . . . 5 ⊢ (¬ y ∈ (F “ B) ↔ ¬ ∃x(x ∈ B ∧ xFy))
49	46, 48	anbi12i 678	. . . 4 ⊢ ((y ∈ (F “ A) ∧ ¬ y ∈ (F “ B)) ↔ (∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy)))
50	45, 49	bitri 240	. . 3 ⊢ (y ∈ ((F “ A) ∖ (F “ B)) ↔ (∃x(x ∈ A ∧ xFy) ∧ ¬ ∃x(x ∈ B ∧ xFy)))
51	39, 44, 50	3bitr4g 279	. 2 ⊢ (Fun ◡F → (y ∈ (F “ (A ∖ B)) ↔ y ∈ ((F “ A) ∖ (F “ B))))
52	51	eqrdv 2351	1 ⊢ (Fun ◡F → (F “ (A ∖ B)) = ((F “ A) ∖ (F “ B)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205   ∖ cdif 3207   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by:  imain  5173  resdif  5307