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Theorem funen1cnv 32585
 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 8595 . . 3 (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2 funrel 6352 . . . . . . . 8 (Fun {𝑝} → Rel {𝑝})
3 vsnid 4559 . . . . . . . 8 𝑝 ∈ {𝑝}
4 elrel 5640 . . . . . . . 8 ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
52, 3, 4sylancl 589 . . . . . . 7 (Fun {𝑝} → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6 sneq 4532 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
762eximi 1837 . . . . . . 7 (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
85, 7syl 17 . . . . . 6 (Fun {𝑝} → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9 funcnvsn 6385 . . . . . . 7 Fun {⟨𝑥, 𝑦⟩}
109gen2 1798 . . . . . 6 𝑥𝑦Fun {⟨𝑥, 𝑦⟩}
11 19.29r2 1876 . . . . . . 7 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → ∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}))
12 cnveq 5713 . . . . . . . . . 10 ({𝑝} = {⟨𝑥, 𝑦⟩} → {𝑝} = {⟨𝑥, 𝑦⟩})
1312funeqd 6357 . . . . . . . . 9 ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun {𝑝} ↔ Fun {⟨𝑥, 𝑦⟩}))
1413biimpar 481 . . . . . . . 8 (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1514exlimivv 1933 . . . . . . 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 589 . . . . 5 (Fun {𝑝} → Fun {𝑝})
1817ax-gen 1797 . . . 4 𝑝(Fun {𝑝} → Fun {𝑝})
19 19.29r 1875 . . . . 5 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})))
20 funeq 6355 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21 cnveq 5713 . . . . . . . . 9 (𝐹 = {𝑝} → 𝐹 = {𝑝})
2221funeqd 6357 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
2320, 22imbi12d 348 . . . . . . 7 (𝐹 = {𝑝} → ((Fun 𝐹 → Fun 𝐹) ↔ (Fun {𝑝} → Fun {𝑝})))
2423biimpar 481 . . . . . 6 ((𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2524exlimiv 1931 . . . . 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 220 . 2 (𝐹 ≈ 1o → (Fun 𝐹 → Fun 𝐹))
2928impcom 411 1 ((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {csn 4522  ⟨cop 4528   class class class wbr 5032  ◡ccnv 5523  Rel wrel 5529  Fun wfun 6329  1oc1o 8105   ≈ cen 8524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-1o 8112  df-en 8528 This theorem is referenced by:  spthcycl  32607
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