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Theorem imainlem 5007
Description: One direction of imain 5008. 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 2329 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2 elin 3153 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 439 . . . . . . . 8 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦))
4 anandir 533 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
53, 4bitri 177 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
65exbii 1512 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
7 19.40 1538 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
86, 7sylbi 118 . . . . 5 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
91, 8sylbi 118 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
10 df-rex 2329 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
11 df-rex 2329 . . . . 5 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
1210, 11anbi12i 441 . . . 4 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
139, 12sylibr 141 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦))
1413ss2abi 3039 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
15 dfima2 4697 . 2 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
16 dfima2 4697 . . . 4 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
17 dfima2 4697 . . . 4 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
1816, 17ineq12i 3163 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
19 inab 3232 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2018, 19eqtri 2076 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2114, 15, 203sstr4i 3011 1 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wa 101  wex 1397  wcel 1409  {cab 2042  wrex 2324  cin 2943  wss 2944   class class class wbr 3791  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  imain  5008
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