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Theorem imainlem 5172
Description: One direction of imain 5173. 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 2397 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2 elin 3227 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 451 . . . . . . . 8 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦))
4 anandir 563 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
53, 4bitri 183 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
65exbii 1567 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)))
7 19.40 1593 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
86, 7sylbi 120 . . . . 5 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
91, 8sylbi 120 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
10 df-rex 2397 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
11 df-rex 2397 . . . . 5 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
1210, 11anbi12i 453 . . . 4 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
139, 12sylibr 133 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦))
1413ss2abi 3137 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
15 dfima2 4851 . 2 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
16 dfima2 4851 . . . 4 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
17 dfima2 4851 . . . 4 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
1816, 17ineq12i 3243 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
19 inab 3312 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2018, 19eqtri 2136 . 2 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑥𝐵 𝑥𝐹𝑦)}
2114, 15, 203sstr4i 3106 1 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wex 1451  wcel 1463  {cab 2101  wrex 2392  cin 3038  wss 3039   class class class wbr 3897  cima 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  imain  5173
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