Theorem funen1cnv 35257

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 𝐹 ∧ 𝐹 ≈ 1o) → Fun ◡𝐹)

Proof of Theorem funen1cnv

Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 8966	. . 3 ⊢ (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2		funrel 6510	. . . . . . . 8 ⊢ (Fun {𝑝} → Rel {𝑝})
3		vsnid 4621	. . . . . . . 8 ⊢ 𝑝 ∈ {𝑝}
4		elrel 5748	. . . . . . . 8 ⊢ ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
5	2, 3, 4	sylancl 587	. . . . . . 7 ⊢ (Fun {𝑝} → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6		sneq 4591	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
7	6	2eximi 1838	. . . . . . 7 ⊢ (∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
8	5, 7	syl 17	. . . . . 6 ⊢ (Fun {𝑝} → ∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩})
9		funcnvsn 6543	. . . . . . 7 ⊢ Fun ◡{⟨𝑥, 𝑦⟩}
10	9	gen2 1798	. . . . . 6 ⊢ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}
11		19.29r2 1877	. . . . . . 7 ⊢ ((∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}) → ∃𝑥∃𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun ◡{⟨𝑥, 𝑦⟩}))
12		cnveq 5823	. . . . . . . . . 10 ⊢ ({𝑝} = {⟨𝑥, 𝑦⟩} → ◡{𝑝} = ◡{⟨𝑥, 𝑦⟩})
13	12	funeqd 6515	. . . . . . . . 9 ⊢ ({𝑝} = {⟨𝑥, 𝑦⟩} → (Fun ◡{𝑝} ↔ Fun ◡{⟨𝑥, 𝑦⟩}))
14	13	biimpar 477	. . . . . . . 8 ⊢ (({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun ◡{⟨𝑥, 𝑦⟩}) → Fun ◡{𝑝})
15	14	exlimivv 1934	. . . . . . 7 ⊢ (∃𝑥∃𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun ◡{⟨𝑥, 𝑦⟩}) → Fun ◡{𝑝})
16	11, 15	syl 17	. . . . . 6 ⊢ ((∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}) → Fun ◡{𝑝})
17	8, 10, 16	sylancl 587	. . . . 5 ⊢ (Fun {𝑝} → Fun ◡{𝑝})
18	17	ax-gen 1797	. . . 4 ⊢ ∀𝑝(Fun {𝑝} → Fun ◡{𝑝})
19		19.29r 1876	. . . . 5 ⊢ ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun ◡{𝑝})) → ∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun ◡{𝑝})))
20		funeq 6513	. . . . . . . 8 ⊢ (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21		cnveq 5823	. . . . . . . . 9 ⊢ (𝐹 = {𝑝} → ◡𝐹 = ◡{𝑝})
22	21	funeqd 6515	. . . . . . . 8 ⊢ (𝐹 = {𝑝} → (Fun ◡𝐹 ↔ Fun ◡{𝑝}))
23	20, 22	imbi12d 344	. . . . . . 7 ⊢ (𝐹 = {𝑝} → ((Fun 𝐹 → Fun ◡𝐹) ↔ (Fun {𝑝} → Fun ◡{𝑝})))
24	23	biimpar 477	. . . . . 6 ⊢ ((𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun ◡{𝑝})) → (Fun 𝐹 → Fun ◡𝐹))
25	24	exlimiv 1932	. . . . 5 ⊢ (∃𝑝(𝐹 = {𝑝} ∧ (Fun {𝑝} → Fun ◡{𝑝})) → (Fun 𝐹 → Fun ◡𝐹))
26	19, 25	syl 17	. . . 4 ⊢ ((∃𝑝 𝐹 = {𝑝} ∧ ∀𝑝(Fun {𝑝} → Fun ◡{𝑝})) → (Fun 𝐹 → Fun ◡𝐹))
27	18, 26	mpan2 692	. . 3 ⊢ (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun ◡𝐹))
28	1, 27	sylbi 217	. 2 ⊢ (𝐹 ≈ 1o → (Fun 𝐹 → Fun ◡𝐹))
29	28	impcom 407	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ≈ 1o) → Fun ◡𝐹)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {csn 4581  ⟨cop 4587   class class class wbr 5099  ^◡ccnv 5624  Rel wrel 5630  Fun wfun 6487  1_oc1o 8393   ≈ cen 8885

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1o 8400  df-en 8889

This theorem is referenced by:  spthcycl  35336