Theorem imadif 5047

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 556	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1537	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1563	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
4	2, 3	sylbi 119	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5		nfv 1462	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 1426	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1498	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 4984	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 2615	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 2615	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 4577	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 1980	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 120	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2021	. . . . . . . . . . . . 13 ⊢ ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	13, 14	sylan 277	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
16	15	con2d 587	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦))
17		imnan 657	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
18	16, 17	sylib 120	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
19	7, 18	alrimi 1456	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 113	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1540	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1429	. . . . . . . 8 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
23	20, 21, 22	3imtr3g 202	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
24	23	anim2d 330	. . . . . 6 ⊢ (Fun ◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
25	4, 24	syl5 32	. . . . 5 ⊢ (Fun ◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))))
26		df-rex 2359	. . . . . 6 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
27		eldif 2993	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
28	27	anbi1i 446	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
29	28	exbii 1537	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
30	26, 29	bitri 182	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
31		df-rex 2359	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
32		df-rex 2359	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
33	32	notbii 627	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
34	31, 33	anbi12i 448	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
35	25, 30, 34	3imtr4g 203	. . . 4 ⊢ (Fun ◡𝐹 → (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)))
36	35	ss2abdv 3078	. . 3 ⊢ (Fun ◡𝐹 → {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)})
37		dfima2 4731	. . 3 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦}
38		dfima2 4731	. . . . 5 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
39		dfima2 4731	. . . . 5 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}
40	38, 39	difeq12i 3100	. . . 4 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦})
41		difab 3251	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
42	40, 41	eqtri 2103	. . 3 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
43	36, 37, 42	3sstr4g 3051	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
44		imadiflem 5046	. . 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 3027	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1283   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ∃*wmo 1944  {cab 2069  ∃wrex 2354   ∖ cdif 2981   ⊆ wss 2984   class class class wbr 3811  ^◡ccnv 4400   “ cima 4404  Fun wfun 4963

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-fun 4971

This theorem is referenced by:  resdif  5223  difpreima  5371  phplem4  6501  phplem4dom  6508  phplem4on  6513