Theorem imadif 6627

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 690	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1881	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1919	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
4	2, 3	sylbi 220	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 2188	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 6559	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 3462	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3462	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 5873	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 2579	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 221	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2656	. . . . . . . . . . . . 13 ⊢ ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	13, 14	sylan 592	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
16	15	con2d 135	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦))
17		imnan 405	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
18	16, 17	sylib 221	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
19	7, 18	alrimi 2252	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 418	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1894	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1814	. . . . . . . 8 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
23	20, 21, 22	3imtr3g 298	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
24	23	anim2d 624	. . . . . 6 ⊢ (Fun ◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
25	4, 24	syl5 35	. . . . 5 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
26		19.29r 1907	. . . . . . 7 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
27	22, 26	sylan2br 607	. . . . . 6 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
28		andi 1025	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29		ianor 997	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦))
30	29	anbi2i 635	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31		an32 659	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵))
32		pm3.24 408	. . . . . . . . . . . 12 ⊢ ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
33	32	intnan 492	. . . . . . . . . . 11 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34		anass 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
35	33, 34	mtbir 326	. . . . . . . . . 10 ⊢ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
36	35	biorfri 953	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
37	31, 36	bitri 278	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
38	28, 30, 37	3bitr4i 306	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
39	38	exbii 1881	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
40	27, 39	sylib 221	. . . . 5 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
41	25, 40	impbid1 228	. . . 4 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
42		eldif 3918	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
43	42	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
44	43	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
45	9	elima2 6073	. . . . 5 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
46	9	elima2 6073	. . . . . 6 ⊢ (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
47	46	notbii 323	. . . . 5 ⊢ (¬ 𝑦 ∈ (𝐹 “ 𝐵) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
48	45, 47	anbi12i 640	. . . 4 ⊢ ((𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
49	41, 44, 48	3bitr4g 317	. . 3 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵))))
50	9	elima2 6073	. . 3 ⊢ (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
51		eldif 3918	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ↔ (𝑦 ∈ (𝐹 “ 𝐴) ∧ ¬ 𝑦 ∈ (𝐹 “ 𝐵)))
52	49, 50, 51	3bitr4g 317	. 2 ⊢ (Fun ◡𝐹 → (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ 𝑦 ∈ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
53	52	eqrdv 2764	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2568   ∖ cdif 3905   class class class wbr 5114  ^◡ccnv 5665   “ cima 5669  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6545

This theorem is used by:  imain  6628  resdif  6849  difpreima  7067  domunsncan  9075  phplem2  9199  php3  9203  infdifsn  9636  cantnfp1lem3  9659  enfin1ai  10386  fin1a2lem7  10408  symgfixelsi  19536  dprdf1o  20135  frlmlbs  21984  f1lindf  22009  cnclima  23462  iscncl  23463  qtopcld  23907  qtoprest  23911  qtopcmap  23913  mbfimaicc  25827  ismbf3d  25850  i1fd  25877  ballotlemfrc  34948  poimirlem2  38313  poimirlem4  38315  poimirlem6  38317  poimirlem7  38318  poimirlem9  38320  poimirlem11  38322  poimirlem12  38323  poimirlem13  38324  poimirlem14  38325  poimirlem16  38327  poimirlem19  38330  poimirlem23  38334