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Theorem imainlem 5289
Description: One direction of imain 5290. 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 2459 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2 elin 3316 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 458 . . . . . . . 8 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦))
4 anandir 591 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
53, 4bitri 184 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
65exbii 1603 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
7 19.40 1629 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
86, 7sylbi 121 . . . . 5 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
91, 8sylbi 121 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
10 df-rex 2459 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
11 df-rex 2459 . . . . 5 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
1210, 11anbi12i 460 . . . 4 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
139, 12sylibr 134 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦))
1413ss2abi 3225 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
15 dfima2 4965 . 2 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
16 dfima2 4965 . . . 4 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
17 dfima2 4965 . . . 4 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
1816, 17ineq12i 3332 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
19 inab 3401 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2018, 19eqtri 2196 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2114, 15, 203sstr4i 3194 1 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wex 1490  wcel 2146  {cab 2161  wrex 2454  cin 3126  wss 3127   class class class wbr 3998  cima 4623
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633
This theorem is referenced by:  imain  5290
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