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Theorem funen1cnv 33692
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 8965 . . 3 (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2 funrel 6518 . . . . . . . 8 (Fun {𝑝} → Rel {𝑝})
3 vsnid 4623 . . . . . . . 8 𝑝 ∈ {𝑝}
4 elrel 5754 . . . . . . . 8 ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
52, 3, 4sylancl 586 . . . . . . 7 (Fun {𝑝} → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6 sneq 4596 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
762eximi 1838 . . . . . . 7 (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
85, 7syl 17 . . . . . 6 (Fun {𝑝} → ∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9 funcnvsn 6551 . . . . . . 7 Fun {⟨𝑥, 𝑦⟩}
109gen2 1798 . . . . . 6 𝑥𝑦Fun {⟨𝑥, 𝑦⟩}
11 19.29r2 1878 . . . . . . 7 ((∃𝑥𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥𝑦Fun {⟨𝑥, 𝑦⟩}) → ∃𝑥𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}))
12 cnveq 5829 . . . . . . . . . 10 ({𝑝} = {⟨𝑥, 𝑦⟩} → {𝑝} = {⟨𝑥, 𝑦⟩})
1312funeqd 6523 . . . . . . . . 9 ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun {𝑝} ↔ Fun {⟨𝑥, 𝑦⟩}))
1413biimpar 478 . . . . . . . 8 (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun {⟨𝑥, 𝑦⟩}) → Fun {𝑝})
1514exlimivv 1935 . . . . . . 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 1797 . . . 4 𝑝(Fun {𝑝} → Fun {𝑝})
19 19.29r 1877 . . . . 5 ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun {𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})))
20 funeq 6521 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21 cnveq 5829 . . . . . . . . 9 (𝐹 = {𝑝} → 𝐹 = {𝑝})
2221funeqd 6523 . . . . . . . 8 (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
2320, 22imbi12d 344 . . . . . . 7 (𝐹 = {𝑝} → ((Fun 𝐹 → Fun 𝐹) ↔ (Fun {𝑝} → Fun {𝑝})))
2423biimpar 478 . . . . . 6 ((𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun {𝑝})) → (Fun 𝐹 → Fun 𝐹))
2524exlimiv 1933 . . . . 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 689 . . 3 (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun 𝐹))
281, 27sylbi 216 . 2 (𝐹 ≈ 1o → (Fun 𝐹 → Fun 𝐹))
2928impcom 408 1 ((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  {csn 4586  cop 4592   class class class wbr 5105  ccnv 5632  Rel wrel 5638  Fun wfun 6490  1oc1o 8405  cen 8880
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1o 8412  df-en 8884
This theorem is referenced by:  spthcycl  33723
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