Theorem imainlem 5000

Description: One direction of imain 5001. This direction does not require Fun ^◡𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)

Assertion

Ref	Expression
imainlem	⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))

Proof of Theorem imainlem

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

Step	Hyp	Ref	Expression
1		df-rex 2354	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
2		elin 3155	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3	2	anbi1i 445	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
4		anandir 555	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5	3, 4	bitri 182	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
6	5	exbii 1536	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
7		19.40 1562	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
8	6, 7	sylbi 119	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
9	1, 8	sylbi 119	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
10		df-rex 2354	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
11		df-rex 2354	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
12	10, 11	anbi12i 447	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
13	9, 12	sylibr 132	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦))
14	13	ss2abi 3066	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
15		dfima2 4690	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}
16		dfima2 4690	. . . 4 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
17		dfima2 4690	. . . 4 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}
18	16, 17	ineq12i 3165	. . 3 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦})
19		inab 3232	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
20	18, 19	eqtri 2101	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
21	14, 15, 20	3sstr4i 3038	1 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))

Syntax hints:   ∧ wa 102  ∃wex 1421   ∈ wcel 1433  {cab 2067  ∃wrex 2349   ∩ cin 2972   ⊆ wss 2973   class class class wbr 3785   “ cima 4366

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  imain  5001