Users' Mathboxes Mathbox for BTernaryTau < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funen1cnv Structured version   Visualization version   GIF version

Theorem funen1cnv 34080
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 9018 . . 3 (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2 funrel 6563 . . . . . . . 8 (Fun {𝑝} → Rel {𝑝})
3 vsnid 4665 . . . . . . . 8 𝑝 ∈ {𝑝}
4 elrel 5797 . . . . . . . 8 ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
52, 3, 4sylancl 587 . . . . . . 7 (Fun {𝑝} → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6 sneq 4638 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
762eximi 1839 . . . . . . 7 (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
85, 7syl 17 . . . . . 6 (Fun {𝑝} → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9 funcnvsn 6596 . . . . . . 7 Fun {⟨𝑥, 𝑦⟩}
109gen2 1799 . . . . . 6 𝑥𝑦Fun {⟨𝑥, 𝑦⟩}
11 19.29r2 1879 . . . . . . 7 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → ∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}))
12 cnveq 5872 . . . . . . . . . 10 ({𝑝} = {⟨𝑥, 𝑦⟩} → {𝑝} = {⟨𝑥, 𝑦⟩})
1312funeqd 6568 . . . . . . . . 9 ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun {𝑝} ↔ Fun {⟨𝑥, 𝑦⟩}))
1413biimpar 479 . . . . . . . 8 (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1514exlimivv 1936 . . . . . . 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 587 . . . . 5 (Fun {𝑝} → Fun {𝑝})
1817ax-gen 1798 . . . 4 𝑝(Fun {𝑝} → Fun {𝑝})
19 19.29r 1878 . . . . 5 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})))
20 funeq 6566 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21 cnveq 5872 . . . . . . . . 9 (𝐹 = {𝑝} → 𝐹 = {𝑝})
2221funeqd 6568 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
2320, 22imbi12d 345 . . . . . . 7 (𝐹 = {𝑝} → ((Fun 𝐹 → Fun 𝐹) ↔ (Fun {𝑝} → Fun {𝑝})))
2423biimpar 479 . . . . . 6 ((𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2524exlimiv 1934 . . . . 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 690 . . 3 (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun 𝐹))
281, 27sylbi 216 . 2 (𝐹 ≈ 1o → (Fun 𝐹 → Fun 𝐹))
2928impcom 409 1 ((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  {csn 4628  cop 4634   class class class wbr 5148  ccnv 5675  Rel wrel 5681  Fun wfun 6535  1oc1o 8456  cen 8933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1o 8463  df-en 8937
This theorem is referenced by:  spthcycl  34109
  Copyright terms: Public domain W3C validator