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 Description: One direction of imadif 5203. This direction does not require Fun ◡𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
Assertion
Ref Expression
imadiflem ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))

Proof of Theorem imadiflem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2422 . . . 4 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
2 df-rex 2422 . . . . 5 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
32notbii 657 . . . 4 (¬ ∃𝑥𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4 alnex 1475 . . . . . . 7 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
5 19.29r 1600 . . . . . . 7 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
64, 5sylan2br 286 . . . . . 6 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
7 simpl 108 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → (𝑥𝐴𝑥𝐹𝑦))
8 simplr 519 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → 𝑥𝐹𝑦)
9 simpr 109 . . . . . . . . . . 11 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
10 ancom 264 . . . . . . . . . . . . 13 ((𝑥𝐵𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦𝑥𝐵))
1110notbii 657 . . . . . . . . . . . 12 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ (𝑥𝐹𝑦𝑥𝐵))
12 imnan 679 . . . . . . . . . . . 12 ((𝑥𝐹𝑦 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐹𝑦𝑥𝐵))
1311, 12bitr4i 186 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
149, 13sylib 121 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
158, 14mpd 13 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → ¬ 𝑥𝐵)
167, 15, 8jca32 308 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
17 eldif 3080 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1817anbi1i 453 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
19 anandir 580 . . . . . . . . 9 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
2018, 19bitri 183 . . . . . . . 8 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
2116, 20sylibr 133 . . . . . . 7 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2221eximi 1579 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
236, 22syl 14 . . . . 5 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
24 df-rex 2422 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2523, 24sylibr 133 . . . 4 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦)
261, 3, 25syl2anb 289 . . 3 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦)
2726ss2abi 3169 . 2 {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
28 dfima2 4883 . . . 4 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
29 dfima2 4883 . . . 4 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
3028, 29difeq12i 3192 . . 3 ((𝐹𝐴) ∖ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
31 difab 3345 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
3230, 31eqtri 2160 . 2 ((𝐹𝐴) ∖ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
33 dfima2 4883 . 2 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
3427, 32, 333sstr4i 3138 1 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329  ∃wex 1468   ∈ wcel 1480  {cab 2125  ∃wrex 2417   ∖ cdif 3068   ⊆ wss 3071   class class class wbr 3929   “ cima 4542 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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552 This theorem is referenced by:  imadif  5203
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