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Theorem map0cor 48685
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 2990 . . . . . 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 8923 . . . . 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 4358 . . . 4 (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴))
1211a1i 11 . . 3 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴)))
13 elmapg 8878 . . . 4 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
1413exbidv 1919 . . 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 1537  wex 1776  wcel 2106  wne 2938  c0 4339  wf 6559  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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