Theorem imain 5245

Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
imain	⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Proof of Theorem imain

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imainlem 5244	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))
2	1	a1i 9	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))
3		eeanv 1909	. . . . . 6 ⊢ (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)))
4		simprll 527	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐴)
5		simpr 109	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥𝐹𝑦)
6		simpr 109	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦) → 𝑧𝐹𝑦)
7	5, 6	anim12i 336	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦))
8		funcnveq 5226	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ◡𝐹 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
9	8	biimpi 119	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡𝐹 → ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
10	9	19.21bi 1535	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡𝐹 → ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
11	10	19.21bbi 1536	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
12	11	imp 123	. . . . . . . . . . . . 13 ⊢ ((Fun ◡𝐹 ∧ (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦)) → 𝑥 = 𝑧)
13	7, 12	sylan2 284	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 = 𝑧)
14		simprrl 529	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑧 ∈ 𝐵)
15	13, 14	eqeltrd 2231	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐵)
16		elin 3286	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
17	4, 15, 16	sylanbrc 414	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴 ∩ 𝐵))
18		simprlr 528	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥𝐹𝑦)
19	17, 18	jca 304	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
20	19	ex 114	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
21	20	exlimdv 1796	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
22	21	eximdv 1857	. . . . . 6 ⊢ (Fun ◡𝐹 → (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
23	3, 22	syl5bir 152	. . . . 5 ⊢ (Fun ◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
24		df-rex 2438	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
25		df-rex 2438	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))
26	24, 25	anbi12i 456	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)))
27		df-rex 2438	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
28	23, 26, 27	3imtr4g 204	. . . 4 ⊢ (Fun ◡𝐹 → ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦))
29	28	ss2abdv 3197	. . 3 ⊢ (Fun ◡𝐹 → {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦})
30		dfima2 4923	. . . . 5 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
31		dfima2 4923	. . . . 5 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}
32	30, 31	ineq12i 3302	. . . 4 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦})
33		inab 3371	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
34	32, 33	eqtri 2175	. . 3 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
35		dfima2 4923	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}
36	29, 34, 35	3sstr4g 3167	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∩ 𝐵)))
37	2, 36	eqssd 3141	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 2125  {cab 2140  ∃wrex 2433   ∩ cin 3097   ⊆ wss 3098   class class class wbr 3961  ^◡ccnv 4578   “ cima 4582  Fun wfun 5157

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-fun 5165

This theorem is referenced by:  inpreima  5586