Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  difmapsn Structured version   Visualization version   GIF version

Theorem difmapsn 42246
 Description: Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
difmapsn.a (𝜑𝐴𝑉)
difmapsn.b (𝜑𝐵𝑊)
difmapsn.v (𝜑𝐶𝑍)
Assertion
Ref Expression
difmapsn (𝜑 → ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) = ((𝐴𝐵) ↑m {𝐶}))

Proof of Theorem difmapsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eldifi 4034 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) → 𝑓 ∈ (𝐴m {𝐶}))
21adantl 485 . . . . . . . . 9 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 ∈ (𝐴m {𝐶}))
3 elmapi 8444 . . . . . . . . . . . 12 (𝑓 ∈ (𝐴m {𝐶}) → 𝑓:{𝐶}⟶𝐴)
43adantl 485 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → 𝑓:{𝐶}⟶𝐴)
5 difmapsn.v . . . . . . . . . . . . 13 (𝜑𝐶𝑍)
6 fsn2g 6897 . . . . . . . . . . . . 13 (𝐶𝑍 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
87adantr 484 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
94, 8mpbid 235 . . . . . . . . . 10 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
109simpld 498 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → (𝑓𝐶) ∈ 𝐴)
112, 10syldan 594 . . . . . . . 8 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓𝐶) ∈ 𝐴)
12 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓𝐶) ∈ 𝐵)
139simprd 499 . . . . . . . . . . . . . 14 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
142, 13syldan 594 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
1514adantr 484 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
1612, 15jca 515 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
17 fsn2g 6897 . . . . . . . . . . . . 13 (𝐶𝑍 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
185, 17syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
1918ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
2016, 19mpbird 260 . . . . . . . . . 10 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓:{𝐶}⟶𝐵)
21 difmapsn.b . . . . . . . . . . . 12 (𝜑𝐵𝑊)
2221ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝐵𝑊)
23 snex 5304 . . . . . . . . . . . 12 {𝐶} ∈ V
2423a1i 11 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → {𝐶} ∈ V)
2522, 24elmapd 8436 . . . . . . . . . 10 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓 ∈ (𝐵m {𝐶}) ↔ 𝑓:{𝐶}⟶𝐵))
2620, 25mpbird 260 . . . . . . . . 9 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓 ∈ (𝐵m {𝐶}))
27 eldifn 4035 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) → ¬ 𝑓 ∈ (𝐵m {𝐶}))
2827ad2antlr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → ¬ 𝑓 ∈ (𝐵m {𝐶}))
2926, 28pm2.65da 816 . . . . . . . 8 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → ¬ (𝑓𝐶) ∈ 𝐵)
3011, 29eldifd 3871 . . . . . . 7 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓𝐶) ∈ (𝐴𝐵))
3130, 14jca 515 . . . . . 6 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
32 fsn2g 6897 . . . . . . . 8 (𝐶𝑍 → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
335, 32syl 17 . . . . . . 7 (𝜑 → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
3433adantr 484 . . . . . 6 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
3531, 34mpbird 260 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓:{𝐶}⟶(𝐴𝐵))
36 difmapsn.a . . . . . . . 8 (𝜑𝐴𝑉)
37 difssd 4040 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
3836, 37ssexd 5198 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
3923a1i 11 . . . . . . 7 (𝜑 → {𝐶} ∈ V)
4038, 39elmapd 8436 . . . . . 6 (𝜑 → (𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴𝐵)))
4140adantr 484 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴𝐵)))
4235, 41mpbird 260 . . . 4 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
4342ralrimiva 3113 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
44 dfss3 3882 . . 3 (((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) ⊆ ((𝐴𝐵) ↑m {𝐶}) ↔ ∀𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
4543, 44sylibr 237 . 2 (𝜑 → ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) ⊆ ((𝐴𝐵) ↑m {𝐶}))
465snn0d 4671 . . 3 (𝜑 → {𝐶} ≠ ∅)
4736, 21, 39, 46difmap 42241 . 2 (𝜑 → ((𝐴𝐵) ↑m {𝐶}) ⊆ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})))
4845, 47eqssd 3911 1 (𝜑 → ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) = ((𝐴𝐵) ↑m {𝐶}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ⊆ wss 3860  {csn 4525  ⟨cop 4531  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ↑m cmap 8422 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-map 8424 This theorem is referenced by:  vonvolmbllem  43700  vonvolmbl  43701
 Copyright terms: Public domain W3C validator