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Theorem f1mo 48762
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 48735 . 2 (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2 f102g 48761 . . 3 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
3 vex 3484 . . . . . . 7 𝑦 ∈ V
4 f1sn2g 48760 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1𝐵)
53, 4mpan 690 . . . . . 6 (𝐹:{𝑦}⟶𝐵𝐹:{𝑦}–1-1𝐵)
6 feq2 6717 . . . . . . 7 (𝐴 = {𝑦} → (𝐹:𝐴𝐵𝐹:{𝑦}⟶𝐵))
7 f1eq2 6800 . . . . . . 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 1930 . . . 4 (∃𝑦 𝐴 = {𝑦} → (𝐹:𝐴𝐵𝐹:𝐴1-1𝐵))
1110imp 406 . . 3 ((∃𝑦 𝐴 = {𝑦} ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
122, 11jaoian 959 . 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 848   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  Vcvv 3480  c0 4333  {csn 4626  wf 6557  1-1wf1 6558
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-pr 5432
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-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  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-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  thincfth  49101
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