Theorem imadif 5203

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 580	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
2	1	exbii 1584	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
3		19.40 1610	. . . . . . 7 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
4	2, 3	sylbi 120	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5		nfv 1508	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Fun ◡𝐹
6		nfe1 1472	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)
7	5, 6	nfan 1544	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵))
8		funmo 5138	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥)
9		vex 2689	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 2689	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
11	9, 10	brcnv 4722	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
12	11	mobii 2036	. . . . . . . . . . . . . 14 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
13	8, 12	sylib 121	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14		mopick 2077	. . . . . . . . . . . . 13 ⊢ ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	13, 14	sylan 281	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
16	15	con2d 613	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦))
17		imnan 679	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
18	16, 17	sylib 121	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
19	7, 18	alrimi 1502	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
20	19	ex 114	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21		exancom 1587	. . . . . . . 8 ⊢ (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
22		alnex 1475	. . . . . . . 8 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
23	20, 21, 22	3imtr3g 203	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
24	23	anim2d 335	. . . . . 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 2422	. . . . . 6 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
27		eldif 3080	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
28	27	anbi1i 453	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
29	28	exbii 1584	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
30	26, 29	bitri 183	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
31		df-rex 2422	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
32		df-rex 2422	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
33	32	notbii 657	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
34	31, 33	anbi12i 455	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
35	25, 30, 34	3imtr4g 204	. . . 4 ⊢ (Fun ◡𝐹 → (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)))
36	35	ss2abdv 3170	. . 3 ⊢ (Fun ◡𝐹 → {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)})
37		dfima2 4883	. . 3 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦}
38		dfima2 4883	. . . . 5 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
39		dfima2 4883	. . . . 5 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}
40	38, 39	difeq12i 3192	. . . 4 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦})
41		difab 3345	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
42	40, 41	eqtri 2160	. . 3 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
43	36, 37, 42	3sstr4g 3140	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
44		imadiflem 5202	. . 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 3114	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃*wmo 2000  {cab 2125  ∃wrex 2417   ∖ cdif 3068   ⊆ wss 3071   class class class wbr 3929  ^◡ccnv 4538   “ cima 4542  Fun wfun 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125

This theorem is referenced by:  resdif  5389  difpreima  5547  phplem4  6749  phplem4dom  6756  phplem4on  6761  cnclima  12392