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Theorem imainlem 4941
Description: One direction of imain 4942. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
Assertion
Ref Expression
imainlem (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))

Proof of Theorem imainlem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2309 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2 elin 3123 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 431 . . . . . . . 8 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦))
4 anandir 525 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
53, 4bitri 173 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
65exbii 1496 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
7 19.40 1522 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
86, 7sylbi 114 . . . . 5 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
91, 8sylbi 114 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
10 df-rex 2309 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
11 df-rex 2309 . . . . 5 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
1210, 11anbi12i 433 . . . 4 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
139, 12sylibr 137 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦))
1413ss2abi 3009 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
15 dfima2 4631 . 2 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
16 dfima2 4631 . . . 4 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
17 dfima2 4631 . . . 4 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
1816, 17ineq12i 3133 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
19 inab 3202 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2018, 19eqtri 2060 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2114, 15, 203sstr4i 2981 1 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wa 97  wex 1381  wcel 1393  {cab 2026  wrex 2304  cin 2913  wss 2914   class class class wbr 3760  cima 4309
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 3871  ax-pow 3923  ax-pr 3940
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 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3761  df-opab 3815  df-xp 4312  df-cnv 4314  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319
This theorem is referenced by:  imain  4942
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