Theorem imadif 6622

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 689	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1878	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1916	. . . . . . 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 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 2185	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 6554	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 3459	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3459	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 5870	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 2576	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 221	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2653	. . . . . . . . . . . . 13 ⊢ ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	13, 14	sylan 591	. . . . . . . . . . . 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 404	. . . . . . . . . . 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 2249	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 417	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1891	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1811	. . . . . . . 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 623	. . . . . 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 1904	. . . . . . 7 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
27	22, 26	sylan2br 606	. . . . . 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 634	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31		an32 658	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥 ∈ 𝐵))
32		pm3.24 407	. . . . . . . . . . . 12 ⊢ ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
33	32	intnan 491	. . . . . . . . . . 11 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34		anass 473	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
35	33, 34	mtbir 326	. . . . . . . . . 10 ⊢ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
36	35	biorfri 952	. . . . . . . . 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 1878	. . . . . 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 3916	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
43	42	anbi1i 635	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
44	43	exbii 1878	. . . 4 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
45	9	elima2 6070	. . . . 5 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
46	9	elima2 6070	. . . . . 6 ⊢ (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
47	46	notbii 323	. . . . 5 ⊢ (¬ 𝑦 ∈ (𝐹 “ 𝐵) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
48	45, 47	anbi12i 639	. . . 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 6070	. . 3 ⊢ (𝑦 ∈ (𝐹 “ (𝐴 ∖ 𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
51		eldif 3916	. . 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 2761	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565   ∖ cdif 3903   class class class wbr 5110  ^◡ccnv 5662   “ cima 5666  Fun wfun 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6540

This theorem is referenced by:  imain  6623  resdif  6844  difpreima  7062  domunsncan  9066  phplem2  9190  php3  9194  infdifsn  9627  cantnfp1lem3  9650  enfin1ai  10369  fin1a2lem7  10391  symgfixelsi  19506  dprdf1o  20105  frlmlbs  21928  f1lindf  21953  cnclima  23406  iscncl  23407  qtopcld  23851  qtoprest  23855  qtopcmap  23857  mbfimaicc  25771  ismbf3d  25794  i1fd  25821  ballotlemfrc  34895  poimirlem2  38251  poimirlem4  38253  poimirlem6  38255  poimirlem7  38256  poimirlem9  38258  poimirlem11  38260  poimirlem12  38261  poimirlem13  38262  poimirlem14  38263  poimirlem16  38265  poimirlem19  38268  poimirlem23  38272