Theorem imadif 6569

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 674	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1849	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1888	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
4	2, 3	sylbi 216	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 2146	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 6500	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 3445	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3445	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 5825	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 2546	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 217	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2625	. . . . . . . . . . . . 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 400	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
18	16, 17	sylib 217	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
19	7, 18	alrimi 2205	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 413	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1863	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1782	. . . . . . . 8 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
23	20, 21, 22	3imtr3g 294	. . . . . . 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 1876	. . . . . . 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 1005	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29		ianor 979	. . . . . . . . 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 643	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵))
32		pm3.24 403	. . . . . . . . . . . 12 ⊢ ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
33	32	intnan 487	. . . . . . . . . . 11 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34		anass 469	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
35	33, 34	mtbir 322	. . . . . . . . . 10 ⊢ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
36	35	biorfi 936	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
37	31, 36	bitri 274	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
38	28, 30, 37	3bitr4i 302	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
39	38	exbii 1849	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
40	27, 39	sylib 217	. . . . 5 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
41	25, 40	impbid1 224	. . . 4 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
42		eldif 3908	. . . . . 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 1849	. . . 4 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
45	9	elima2 6006	. . . . 5 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
46	9	elima2 6006	. . . . . 6 ⊢ (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
47	46	notbii 319	. . . . 5 ⊢ (¬ 𝑦 ∈ (𝐹 “ 𝐵) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
48	45, 47	anbi12i 627	. . . 4 ⊢ ((𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
49	41, 44, 48	3bitr4g 313	. . 3 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵))))
50	9	elima2 6006	. . 3 ⊢ (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
51		eldif 3908	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)))
52	49, 50, 51	3bitr4g 313	. 2 ⊢ (Fun ◡𝐹 → (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ 𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
53	52	eqrdv 2734	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃*wmo 2536   ∖ cdif 3895   class class class wbr 5093  ^◡ccnv 5620   “ cima 5624  Fun wfun 6474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-br 5094  df-opab 5156  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6482

This theorem is referenced by:  imain  6570  resdif  6789  difpreima  6999  domunsncan  8938  phplem2  9074  php3  9078  phplem4OLD  9086  php3OLD  9090  infdifsn  9515  cantnfp1lem3  9538  enfin1ai  10242  fin1a2lem7  10264  symgfixelsi  19140  dprdf1o  19731  frlmlbs  21111  f1lindf  21136  cnclima  22526  iscncl  22527  qtopcld  22971  qtoprest  22975  qtopcmap  22977  mbfimaicc  24902  ismbf3d  24925  i1fd  24952  ballotlemfrc  32793  poimirlem2  35935  poimirlem4  35937  poimirlem6  35939  poimirlem7  35940  poimirlem9  35942  poimirlem11  35944  poimirlem12  35945  poimirlem13  35946  poimirlem14  35947  poimirlem16  35949  poimirlem19  35952  poimirlem23  35956