Theorem imadif 6570

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 677	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1848	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1886	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
4	2, 3	sylbi 217	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 2151	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 6502	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 3442	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3442	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 5829	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 2541	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 218	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2618	. . . . . . . . . . . . 13 ⊢ ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	13, 14	sylan 580	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
16	15	con2d 134	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦))
17		imnan 399	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
18	16, 17	sylib 218	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
19	7, 18	alrimi 2214	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 412	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1861	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1781	. . . . . . . 8 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
23	20, 21, 22	3imtr3g 295	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
24	23	anim2d 612	. . . . . 6 ⊢ (Fun ◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
25	4, 24	syl5 34	. . . . 5 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
26		19.29r 1874	. . . . . . 7 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
27	22, 26	sylan2br 595	. . . . . 6 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
28		andi 1009	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29		ianor 983	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦))
30	29	anbi2i 623	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31		an32 646	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵))
32		pm3.24 402	. . . . . . . . . . . 12 ⊢ ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
33	32	intnan 486	. . . . . . . . . . 11 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34		anass 468	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
35	33, 34	mtbir 323	. . . . . . . . . 10 ⊢ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
36	35	biorfri 939	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
37	31, 36	bitri 275	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
38	28, 30, 37	3bitr4i 303	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
39	38	exbii 1848	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
40	27, 39	sylib 218	. . . . 5 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
41	25, 40	impbid1 225	. . . 4 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
42		eldif 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
43	42	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
44	43	exbii 1848	. . . 4 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
45	9	elima2 6021	. . . . 5 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
46	9	elima2 6021	. . . . . 6 ⊢ (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
47	46	notbii 320	. . . . 5 ⊢ (¬ 𝑦 ∈ (𝐹 “ 𝐵) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
48	45, 47	anbi12i 628	. . . 4 ⊢ ((𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
49	41, 44, 48	3bitr4g 314	. . 3 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵))))
50	9	elima2 6021	. . 3 ⊢ (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
51		eldif 3915	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)))
52	49, 50, 51	3bitr4g 314	. 2 ⊢ (Fun ◡𝐹 → (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ 𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
53	52	eqrdv 2727	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531   ∖ cdif 3902   class class class wbr 5095  ^◡ccnv 5622   “ cima 5626  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488

This theorem is referenced by:  imain  6571  f1imadifssran  6572  resdif  6789  difpreima  7003  domunsncan  9001  phplem2  9129  php3  9133  infdifsn  9572  cantnfp1lem3  9595  enfin1ai  10297  fin1a2lem7  10319  symgfixelsi  19332  dprdf1o  19931  frlmlbs  21722  f1lindf  21747  cnclima  23171  iscncl  23172  qtopcld  23616  qtoprest  23620  qtopcmap  23622  mbfimaicc  25548  ismbf3d  25571  i1fd  25598  ballotlemfrc  34494  poimirlem2  37601  poimirlem4  37603  poimirlem6  37605  poimirlem7  37606  poimirlem9  37608  poimirlem11  37610  poimirlem12  37611  poimirlem13  37612  poimirlem14  37613  poimirlem16  37615  poimirlem19  37618  poimirlem23  37622