Theorem imain 5412

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 5411	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))
2	1	a1i 9	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))
3		eeanv 1985	. . . . . 6 ⊢ (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)))
4		simprll 539	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐴)
5		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥𝐹𝑦)
6		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦) → 𝑧𝐹𝑦)
7	5, 6	anim12i 338	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦))
8		funcnveq 5393	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ◡𝐹 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
9	8	biimpi 120	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡𝐹 → ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
10	9	19.21bi 1606	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡𝐹 → ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
11	10	19.21bbi 1607	. . . . . . . . . . . . . 14 ⊢ (Fun ◡𝐹 → ((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
12	11	imp 124	. . . . . . . . . . . . 13 ⊢ ((Fun ◡𝐹 ∧ (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦)) → 𝑥 = 𝑧)
13	7, 12	sylan2 286	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 = 𝑧)
14		simprrl 541	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑧 ∈ 𝐵)
15	13, 14	eqeltrd 2308	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐵)
16		elin 3390	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
17	4, 15, 16	sylanbrc 417	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴 ∩ 𝐵))
18		simprlr 540	. . . . . . . . . 10 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥𝐹𝑦)
19	17, 18	jca 306	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
20	19	ex 115	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
21	20	exlimdv 1867	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
22	21	eximdv 1928	. . . . . 6 ⊢ (Fun ◡𝐹 → (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
23	3, 22	biimtrrid 153	. . . . 5 ⊢ (Fun ◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
24		df-rex 2516	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
25		df-rex 2516	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))
26	24, 25	anbi12i 460	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)))
27		df-rex 2516	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
28	23, 26, 27	3imtr4g 205	. . . 4 ⊢ (Fun ◡𝐹 → ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦))
29	28	ss2abdv 3300	. . 3 ⊢ (Fun ◡𝐹 → {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦})
30		dfima2 5078	. . . . 5 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
31		dfima2 5078	. . . . 5 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}
32	30, 31	ineq12i 3406	. . . 4 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦})
33		inab 3475	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
34	32, 33	eqtri 2252	. . 3 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
35		dfima2 5078	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}
36	29, 34, 35	3sstr4g 3270	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∩ 𝐵)))
37	2, 36	eqssd 3244	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  ∃wrex 2511   ∩ cin 3199   ⊆ wss 3200   class class class wbr 4088  ^◡ccnv 4724   “ cima 4728  Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328

This theorem is referenced by:  inpreima  5773