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Theorem funen1cnv 35102
Description: If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
Assertion
Ref Expression
funen1cnv ((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)

Proof of Theorem funen1cnv
Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 en1 9064 . . 3 (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2 funrel 6583 . . . . . . . 8 (Fun {𝑝} → Rel {𝑝})
3 vsnid 4663 . . . . . . . 8 𝑝 ∈ {𝑝}
4 elrel 5808 . . . . . . . 8 ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
52, 3, 4sylancl 586 . . . . . . 7 (Fun {𝑝} → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6 sneq 4636 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
762eximi 1836 . . . . . . 7 (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
85, 7syl 17 . . . . . 6 (Fun {𝑝} → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9 funcnvsn 6616 . . . . . . 7 Fun {⟨𝑥, 𝑦⟩}
109gen2 1796 . . . . . 6 𝑥𝑦Fun {⟨𝑥, 𝑦⟩}
11 19.29r2 1875 . . . . . . 7 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → ∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}))
12 cnveq 5884 . . . . . . . . . 10 ({𝑝} = {⟨𝑥, 𝑦⟩} → {𝑝} = {⟨𝑥, 𝑦⟩})
1312funeqd 6588 . . . . . . . . 9 ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun {𝑝} ↔ Fun {⟨𝑥, 𝑦⟩}))
1413biimpar 477 . . . . . . . 8 (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1514exlimivv 1932 . . . . . . 7 (∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1611, 15syl 17 . . . . . 6 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
178, 10, 16sylancl 586 . . . . 5 (Fun {𝑝} → Fun {𝑝})
1817ax-gen 1795 . . . 4 𝑝(Fun {𝑝} → Fun {𝑝})
19 19.29r 1874 . . . . 5 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})))
20 funeq 6586 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21 cnveq 5884 . . . . . . . . 9 (𝐹 = {𝑝} → 𝐹 = {𝑝})
2221funeqd 6588 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
2320, 22imbi12d 344 . . . . . . 7 (𝐹 = {𝑝} → ((Fun 𝐹 → Fun 𝐹) ↔ (Fun {𝑝} → Fun {𝑝})))
2423biimpar 477 . . . . . 6 ((𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2524exlimiv 1930 . . . . 5 (∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2619, 25syl 17 . . . 4 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2718, 26mpan2 691 . . 3 (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun 𝐹))
281, 27sylbi 217 . 2 (𝐹 ≈ 1o → (Fun 𝐹 → Fun 𝐹))
2928impcom 407 1 ((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {csn 4626  cop 4632   class class class wbr 5143  ccnv 5684  Rel wrel 5690  Fun wfun 6555  1oc1o 8499  cen 8982
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-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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  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-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986
This theorem is referenced by:  spthcycl  35134
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