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Theorem map0cor 48764
Description: A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
Hypotheses
Ref Expression
map0cor.1 (𝜑𝐴𝑉)
map0cor.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
map0cor (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑉   𝑓,𝑊
Allowed substitution hint:   𝜑(𝑓)

Proof of Theorem map0cor
StepHypRef Expression
1 map0cor.2 . 2 (𝜑𝐵𝑊)
2 map0cor.1 . 2 (𝜑𝐴𝑉)
3 biid 261 . . . . . . 7 (𝐴 ≠ ∅ ↔ 𝐴 ≠ ∅)
43necon2bbii 2992 . . . . . 6 (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)
54imbi2i 336 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐵 = ∅ → ¬ 𝐴 ≠ ∅))
6 imnan 399 . . . . 5 ((𝐵 = ∅ → ¬ 𝐴 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
75, 6bitri 275 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
8 map0g 8924 . . . . 5 ((𝐵𝑊𝐴𝑉) → ((𝐵m 𝐴) = ∅ ↔ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)))
98notbid 318 . . . 4 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)))
107, 9bitr4id 290 . . 3 ((𝐵𝑊𝐴𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵m 𝐴) = ∅))
11 neq0 4352 . . . 4 (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴))
1211a1i 11 . . 3 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴)))
13 elmapg 8879 . . . 4 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
1413exbidv 1921 . . 3 ((𝐵𝑊𝐴𝑉) → (∃𝑓 𝑓 ∈ (𝐵m 𝐴) ↔ ∃𝑓 𝑓:𝐴𝐵))
1510, 12, 143bitrd 305 . 2 ((𝐵𝑊𝐴𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴𝐵))
161, 2, 15syl2anc 584 1 (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  c0 4333  wf 6557  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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