Theorem imainlem 4980

Description: One direction of imain 4981. 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 2312	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))
2		elin 3126	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3	2	anbi1i 431	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
4		anandir 525	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
5	3, 4	bitri 173	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
6	5	exbii 1496	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
7		19.40 1522	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
8	6, 7	sylbi 114	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
9	1, 8	sylbi 114	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
10		df-rex 2312	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
11		df-rex 2312	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
12	10, 11	anbi12i 433	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
13	9, 12	sylibr 137	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦))
14	13	ss2abi 3012	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
15		dfima2 4670	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}
16		dfima2 4670	. . . 4 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
17		dfima2 4670	. . . 4 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}
18	16, 17	ineq12i 3136	. . 3 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦})
19		inab 3205	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
20	18, 19	eqtri 2060	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
21	14, 15, 20	3sstr4i 2984	1 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))

Syntax hints:   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2307   ∩ cin 2916   ⊆ wss 2917   class class class wbr 3764   “ cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by:  imain  4981