Theorem imadif 4920

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 525	. . . . . . . 8 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y) ↔ ((x ∈ A ∧ x𝐹y) ∧ (¬ x ∈ B ∧ x𝐹y)))
2	1	exbii 1493	. . . . . . 7 ⊢ (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y) ↔ ∃x((x ∈ A ∧ x𝐹y) ∧ (¬ x ∈ B ∧ x𝐹y)))
3		19.40 1519	. . . . . . 7 ⊢ (∃x((x ∈ A ∧ x𝐹y) ∧ (¬ x ∈ B ∧ x𝐹y)) → (∃x(x ∈ A ∧ x𝐹y) ∧ ∃x(¬ x ∈ B ∧ x𝐹y)))
4	2, 3	sylbi 114	. . . . . 6 ⊢ (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y) → (∃x(x ∈ A ∧ x𝐹y) ∧ ∃x(¬ x ∈ B ∧ x𝐹y)))
5		nfv 1418	. . . . . . . . . . 11 ⊢ ℲxFun ◡𝐹
6		nfe1 1382	. . . . . . . . . . 11 ⊢ Ⅎx∃x(x𝐹y ∧ ¬ x ∈ B)
7	5, 6	nfan 1454	. . . . . . . . . 10 ⊢ Ⅎx(Fun ◡𝐹 ∧ ∃x(x𝐹y ∧ ¬ x ∈ B))
8		funmo 4858	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*x y◡𝐹x)
9		vex 2554	. . . . . . . . . . . . . . . 16 ⊢ y ∈ V
10		vex 2554	. . . . . . . . . . . . . . . 16 ⊢ x ∈ V
11	9, 10	brcnv 4460	. . . . . . . . . . . . . . 15 ⊢ (y◡𝐹x ↔ x𝐹y)
12	11	mobii 1934	. . . . . . . . . . . . . 14 ⊢ (∃*x y◡𝐹x ↔ ∃*x x𝐹y)
13	8, 12	sylib 127	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*x x𝐹y)
14		mopick 1975	. . . . . . . . . . . . 13 ⊢ ((∃*x x𝐹y ∧ ∃x(x𝐹y ∧ ¬ x ∈ B)) → (x𝐹y → ¬ x ∈ B))
15	13, 14	sylan 267	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ∃x(x𝐹y ∧ ¬ x ∈ B)) → (x𝐹y → ¬ x ∈ B))
16	15	con2d 554	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ∃x(x𝐹y ∧ ¬ x ∈ B)) → (x ∈ B → ¬ x𝐹y))
17		imnan 623	. . . . . . . . . . 11 ⊢ ((x ∈ B → ¬ x𝐹y) ↔ ¬ (x ∈ B ∧ x𝐹y))
18	16, 17	sylib 127	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃x(x𝐹y ∧ ¬ x ∈ B)) → ¬ (x ∈ B ∧ x𝐹y))
19	7, 18	alrimi 1412	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃x(x𝐹y ∧ ¬ x ∈ B)) → ∀x ¬ (x ∈ B ∧ x𝐹y))
20	19	ex 108	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃x(x𝐹y ∧ ¬ x ∈ B) → ∀x ¬ (x ∈ B ∧ x𝐹y)))
21		exancom 1496	. . . . . . . 8 ⊢ (∃x(x𝐹y ∧ ¬ x ∈ B) ↔ ∃x(¬ x ∈ B ∧ x𝐹y))
22		alnex 1385	. . . . . . . 8 ⊢ (∀x ¬ (x ∈ B ∧ x𝐹y) ↔ ¬ ∃x(x ∈ B ∧ x𝐹y))
23	20, 21, 22	3imtr3g 193	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃x(¬ x ∈ B ∧ x𝐹y) → ¬ ∃x(x ∈ B ∧ x𝐹y)))
24	23	anim2d 320	. . . . . 6 ⊢ (Fun ◡𝐹 → ((∃x(x ∈ A ∧ x𝐹y) ∧ ∃x(¬ x ∈ B ∧ x𝐹y)) → (∃x(x ∈ A ∧ x𝐹y) ∧ ¬ ∃x(x ∈ B ∧ x𝐹y))))
25	4, 24	syl5 28	. . . . 5 ⊢ (Fun ◡𝐹 → (∃x((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y) → (∃x(x ∈ A ∧ x𝐹y) ∧ ¬ ∃x(x ∈ B ∧ x𝐹y))))
26		df-rex 2306	. . . . . 6 ⊢ (∃x ∈ (A ∖ B)x𝐹y ↔ ∃x(x ∈ (A ∖ B) ∧ x𝐹y))
27		eldif 2921	. . . . . . . 8 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
28	27	anbi1i 431	. . . . . . 7 ⊢ ((x ∈ (A ∖ B) ∧ x𝐹y) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y))
29	28	exbii 1493	. . . . . 6 ⊢ (∃x(x ∈ (A ∖ B) ∧ x𝐹y) ↔ ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y))
30	26, 29	bitri 173	. . . . 5 ⊢ (∃x ∈ (A ∖ B)x𝐹y ↔ ∃x((x ∈ A ∧ ¬ x ∈ B) ∧ x𝐹y))
31		df-rex 2306	. . . . . 6 ⊢ (∃x ∈ A x𝐹y ↔ ∃x(x ∈ A ∧ x𝐹y))
32		df-rex 2306	. . . . . . 7 ⊢ (∃x ∈ B x𝐹y ↔ ∃x(x ∈ B ∧ x𝐹y))
33	32	notbii 593	. . . . . 6 ⊢ (¬ ∃x ∈ B x𝐹y ↔ ¬ ∃x(x ∈ B ∧ x𝐹y))
34	31, 33	anbi12i 433	. . . . 5 ⊢ ((∃x ∈ A x𝐹y ∧ ¬ ∃x ∈ B x𝐹y) ↔ (∃x(x ∈ A ∧ x𝐹y) ∧ ¬ ∃x(x ∈ B ∧ x𝐹y)))
35	25, 30, 34	3imtr4g 194	. . . 4 ⊢ (Fun ◡𝐹 → (∃x ∈ (A ∖ B)x𝐹y → (∃x ∈ A x𝐹y ∧ ¬ ∃x ∈ B x𝐹y)))
36	35	ss2abdv 3007	. . 3 ⊢ (Fun ◡𝐹 → {y ∣ ∃x ∈ (A ∖ B)x𝐹y} ⊆ {y ∣ (∃x ∈ A x𝐹y ∧ ¬ ∃x ∈ B x𝐹y)})
37		dfima2 4612	. . 3 ⊢ (𝐹 “ (A ∖ B)) = {y ∣ ∃x ∈ (A ∖ B)x𝐹y}
38		dfima2 4612	. . . . 5 ⊢ (𝐹 “ A) = {y ∣ ∃x ∈ A x𝐹y}
39		dfima2 4612	. . . . 5 ⊢ (𝐹 “ B) = {y ∣ ∃x ∈ B x𝐹y}
40	38, 39	difeq12i 3054	. . . 4 ⊢ ((𝐹 “ A) ∖ (𝐹 “ B)) = ({y ∣ ∃x ∈ A x𝐹y} ∖ {y ∣ ∃x ∈ B x𝐹y})
41		difab 3200	. . . 4 ⊢ ({y ∣ ∃x ∈ A x𝐹y} ∖ {y ∣ ∃x ∈ B x𝐹y}) = {y ∣ (∃x ∈ A x𝐹y ∧ ¬ ∃x ∈ B x𝐹y)}
42	40, 41	eqtri 2057	. . 3 ⊢ ((𝐹 “ A) ∖ (𝐹 “ B)) = {y ∣ (∃x ∈ A x𝐹y ∧ ¬ ∃x ∈ B x𝐹y)}
43	36, 37, 42	3sstr4g 2980	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (A ∖ B)) ⊆ ((𝐹 “ A) ∖ (𝐹 “ B)))
44		imadiflem 4919	. . 3 ⊢ ((𝐹 “ A) ∖ (𝐹 “ B)) ⊆ (𝐹 “ (A ∖ B))
45	44	a1i 9	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ A) ∖ (𝐹 “ B)) ⊆ (𝐹 “ (A ∖ B)))
46	43, 45	eqssd 2956	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (A ∖ B)) = ((𝐹 “ A) ∖ (𝐹 “ B)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  {cab 2023  ∃wrex 2301   ∖ cdif 2908   ⊆ wss 2911   class class class wbr 3754  ^◡ccnv 4286   “ cima 4290  Fun wfun 4838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-fun 4846

This theorem is referenced by:  resdif  5089  difpreima  5235