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Theorem funen1cnv 34546
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 9016 . . 3 (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2 funrel 6555 . . . . . . . 8 (Fun {𝑝} → Rel {𝑝})
3 vsnid 4657 . . . . . . . 8 𝑝 ∈ {𝑝}
4 elrel 5788 . . . . . . . 8 ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
52, 3, 4sylancl 585 . . . . . . 7 (Fun {𝑝} → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6 sneq 4630 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
762eximi 1830 . . . . . . 7 (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
85, 7syl 17 . . . . . 6 (Fun {𝑝} → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9 funcnvsn 6588 . . . . . . 7 Fun {⟨𝑥, 𝑦⟩}
109gen2 1790 . . . . . 6 𝑥𝑦Fun {⟨𝑥, 𝑦⟩}
11 19.29r2 1870 . . . . . . 7 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → ∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}))
12 cnveq 5863 . . . . . . . . . 10 ({𝑝} = {⟨𝑥, 𝑦⟩} → {𝑝} = {⟨𝑥, 𝑦⟩})
1312funeqd 6560 . . . . . . . . 9 ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun {𝑝} ↔ Fun {⟨𝑥, 𝑦⟩}))
1413biimpar 477 . . . . . . . 8 (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1514exlimivv 1927 . . . . . . 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 585 . . . . 5 (Fun {𝑝} → Fun {𝑝})
1817ax-gen 1789 . . . 4 𝑝(Fun {𝑝} → Fun {𝑝})
19 19.29r 1869 . . . . 5 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})))
20 funeq 6558 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21 cnveq 5863 . . . . . . . . 9 (𝐹 = {𝑝} → 𝐹 = {𝑝})
2221funeqd 6560 . . . . . . . 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 1925 . . . . 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 688 . . 3 (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun 𝐹))
281, 27sylbi 216 . 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 1531   = wceq 1533  wex 1773  wcel 2098  {csn 4620  cop 4626   class class class wbr 5138  ccnv 5665  Rel wrel 5671  Fun wfun 6527  1oc1o 8454  cen 8931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-1o 8461  df-en 8935
This theorem is referenced by:  spthcycl  34575
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