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Theorem map0cor 46139
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 260 . . . . . . 7 (𝐴 ≠ ∅ ↔ 𝐴 ≠ ∅)
43necon2bbii 2995 . . . . . 6 (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)
54imbi2i 336 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐵 = ∅ → ¬ 𝐴 ≠ ∅))
6 imnan 400 . . . . 5 ((𝐵 = ∅ → ¬ 𝐴 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
75, 6bitri 274 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
8 map0g 8661 . . . . 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 4281 . . . 4 (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴))
1211a1i 11 . . 3 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴)))
13 elmapg 8617 . . . 4 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
1413exbidv 1924 . . 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 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wne 2943  c0 4258  wf 6424  (class class class)co 7269  m cmap 8604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-fv 6436  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-map 8606
This theorem is referenced by: (None)
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