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Theorem f1mo 48683
Description: A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
Assertion
Ref Expression
f1mo ((∃*𝑥 𝑥𝐴𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem f1mo
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mo0sn 48664 . 2 (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2 f102g 48682 . . 3 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
3 vex 3482 . . . . . . 7 𝑦 ∈ V
4 f1sn2g 48681 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1𝐵)
53, 4mpan 690 . . . . . 6 (𝐹:{𝑦}⟶𝐵𝐹:{𝑦}–1-1𝐵)
6 feq2 6718 . . . . . . 7 (𝐴 = {𝑦} → (𝐹:𝐴𝐵𝐹:{𝑦}⟶𝐵))
7 f1eq2 6801 . . . . . . 7 (𝐴 = {𝑦} → (𝐹:𝐴1-1𝐵𝐹:{𝑦}–1-1𝐵))
86, 7imbi12d 344 . . . . . 6 (𝐴 = {𝑦} → ((𝐹:𝐴𝐵𝐹:𝐴1-1𝐵) ↔ (𝐹:{𝑦}⟶𝐵𝐹:{𝑦}–1-1𝐵)))
95, 8mpbiri 258 . . . . 5 (𝐴 = {𝑦} → (𝐹:𝐴𝐵𝐹:𝐴1-1𝐵))
109exlimiv 1928 . . . 4 (∃𝑦 𝐴 = {𝑦} → (𝐹:𝐴𝐵𝐹:𝐴1-1𝐵))
1110imp 406 . . 3 ((∃𝑦 𝐴 = {𝑦} ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
122, 11jaoian 958 . 2 (((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
131, 12sylanb 581 1 ((∃*𝑥 𝑥𝐴𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  ∃*wmo 2536  Vcvv 3478  c0 4339  {csn 4631  wf 6559  1-1wf1 6560
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-pr 5438
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-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  thincfth  48848
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