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Theorem map0cor 46222
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 2990 . . . . . 6 (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)
54imbi2i 335 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐵 = ∅ → ¬ 𝐴 ≠ ∅))
6 imnan 399 . . . . 5 ((𝐵 = ∅ → ¬ 𝐴 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
75, 6bitri 274 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))
8 map0g 8692 . . . . 5 ((𝐵𝑊𝐴𝑉) → ((𝐵m 𝐴) = ∅ ↔ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)))
98notbid 317 . . . 4 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)))
107, 9bitr4id 289 . . 3 ((𝐵𝑊𝐴𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵m 𝐴) = ∅))
11 neq0 4282 . . . 4 (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴))
1211a1i 11 . . 3 ((𝐵𝑊𝐴𝑉) → (¬ (𝐵m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵m 𝐴)))
13 elmapg 8648 . . . 4 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
1413exbidv 1920 . . 3 ((𝐵𝑊𝐴𝑉) → (∃𝑓 𝑓 ∈ (𝐵m 𝐴) ↔ ∃𝑓 𝑓:𝐴𝐵))
1510, 12, 143bitrd 304 . 2 ((𝐵𝑊𝐴𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴𝐵))
161, 2, 15syl2anc 583 1 (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1537  wex 1777  wcel 2101  wne 2938  c0 4259  wf 6443  (class class class)co 7295  m cmap 8635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-map 8637
This theorem is referenced by: (None)
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