Theorem imain 5365

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 5364	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))
2	1	a1i 9	. 2 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))
3		eeanv 1961	. . . . . 6 ⊢ (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)))
4		simprll 537	. . . . . . . . . . 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 5346	. . . . . . . . . . . . . . . . 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 1582	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡𝐹 → ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧))
11	10	19.21bbi 1583	. . . . . . . . . . . . . 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 539	. . . . . . . . . . . 12 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑧 ∈ 𝐵)
15	13, 14	eqeltrd 2283	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐵)
16		elin 3360	. . . . . . . . . . 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 538	. . . . . . . . . 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 1843	. . . . . . 7 ⊢ (Fun ◡𝐹 → (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)))
22	21	eximdv 1904	. . . . . 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 2491	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
25		df-rex 2491	. . . . . 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 2491	. . . . 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 3270	. . 3 ⊢ (Fun ◡𝐹 → {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦})
30		dfima2 5033	. . . . 5 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
31		dfima2 5033	. . . . 5 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}
32	30, 31	ineq12i 3376	. . . 4 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦})
33		inab 3445	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
34	32, 33	eqtri 2227	. . 3 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)}
35		dfima2 5033	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}
36	29, 34, 35	3sstr4g 3240	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∩ 𝐵)))
37	2, 36	eqssd 3214	1 ⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {cab 2192  ∃wrex 2486   ∩ cin 3169   ⊆ wss 3170   class class class wbr 4051  ^◡ccnv 4682   “ cima 4686  Fun wfun 5274

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-fun 5282

This theorem is referenced by:  inpreima  5719