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Theorem difmapsn 41351
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 4100 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) → 𝑓 ∈ (𝐴m {𝐶}))
21adantl 482 . . . . . . . . 9 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 ∈ (𝐴m {𝐶}))
3 elmapi 8417 . . . . . . . . . . . 12 (𝑓 ∈ (𝐴m {𝐶}) → 𝑓:{𝐶}⟶𝐴)
43adantl 482 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → 𝑓:{𝐶}⟶𝐴)
5 difmapsn.v . . . . . . . . . . . . 13 (𝜑𝐶𝑍)
6 fsn2g 6892 . . . . . . . . . . . . 13 (𝐶𝑍 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
87adantr 481 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
94, 8mpbid 233 . . . . . . . . . 10 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → ((𝑓𝐶) ∈ 𝐴𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
109simpld 495 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → (𝑓𝐶) ∈ 𝐴)
112, 10syldan 591 . . . . . . . 8 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓𝐶) ∈ 𝐴)
12 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓𝐶) ∈ 𝐵)
139simprd 496 . . . . . . . . . . . . . 14 ((𝜑𝑓 ∈ (𝐴m {𝐶})) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
142, 13syldan 591 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
1514adantr 481 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})
1612, 15jca 512 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
17 fsn2g 6892 . . . . . . . . . . . . 13 (𝐶𝑍 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
185, 17syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
1918ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓𝐶) ∈ 𝐵𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
2016, 19mpbird 258 . . . . . . . . . 10 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓:{𝐶}⟶𝐵)
21 difmapsn.b . . . . . . . . . . . 12 (𝜑𝐵𝑊)
2221ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝐵𝑊)
23 snex 5322 . . . . . . . . . . . 12 {𝐶} ∈ V
2423a1i 11 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → {𝐶} ∈ V)
2522, 24elmapd 8409 . . . . . . . . . 10 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → (𝑓 ∈ (𝐵m {𝐶}) ↔ 𝑓:{𝐶}⟶𝐵))
2620, 25mpbird 258 . . . . . . . . 9 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → 𝑓 ∈ (𝐵m {𝐶}))
27 eldifn 4101 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) → ¬ 𝑓 ∈ (𝐵m {𝐶}))
2827ad2antlr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) ∧ (𝑓𝐶) ∈ 𝐵) → ¬ 𝑓 ∈ (𝐵m {𝐶}))
2926, 28pm2.65da 813 . . . . . . . 8 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → ¬ (𝑓𝐶) ∈ 𝐵)
3011, 29eldifd 3944 . . . . . . 7 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓𝐶) ∈ (𝐴𝐵))
3130, 14jca 512 . . . . . 6 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩}))
32 fsn2g 6892 . . . . . . . 8 (𝐶𝑍 → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
335, 32syl 17 . . . . . . 7 (𝜑 → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
3433adantr 481 . . . . . 6 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓:{𝐶}⟶(𝐴𝐵) ↔ ((𝑓𝐶) ∈ (𝐴𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓𝐶)⟩})))
3531, 34mpbird 258 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓:{𝐶}⟶(𝐴𝐵))
36 difmapsn.a . . . . . . . 8 (𝜑𝐴𝑉)
37 difssd 4106 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
3836, 37ssexd 5219 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
3923a1i 11 . . . . . . 7 (𝜑 → {𝐶} ∈ V)
4038, 39elmapd 8409 . . . . . 6 (𝜑 → (𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴𝐵)))
4140adantr 481 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → (𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴𝐵)))
4235, 41mpbird 258 . . . 4 ((𝜑𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))) → 𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
4342ralrimiva 3179 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
44 dfss3 3953 . . 3 (((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) ⊆ ((𝐴𝐵) ↑m {𝐶}) ↔ ∀𝑓 ∈ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶}))𝑓 ∈ ((𝐴𝐵) ↑m {𝐶}))
4543, 44sylibr 235 . 2 (𝜑 → ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) ⊆ ((𝐴𝐵) ↑m {𝐶}))
465snn0d 41226 . . 3 (𝜑 → {𝐶} ≠ ∅)
4736, 21, 39, 46difmap 41346 . 2 (𝜑 → ((𝐴𝐵) ↑m {𝐶}) ⊆ ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})))
4845, 47eqssd 3981 1 (𝜑 → ((𝐴m {𝐶}) ∖ (𝐵m {𝐶})) = ((𝐴𝐵) ↑m {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  {csn 4557  cop 4563  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by:  vonvolmbllem  42819  vonvolmbl  42820
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