Theorem funen1cnv 35077

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 9046	. . 3 ⊢ (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2		funrel 6563	. . . . . . . 8 ⊢ (Fun {𝑝} → Rel {𝑝})
3		vsnid 4643	. . . . . . . 8 ⊢ 𝑝 ∈ {𝑝}
4		elrel 5788	. . . . . . . 8 ⊢ ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
5	2, 3, 4	sylancl 586	. . . . . . 7 ⊢ (Fun {𝑝} → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6		sneq 4616	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
7	6	2eximi 1835	. . . . . . 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 6596	. . . . . . 7 ⊢ Fun ◡{⟨𝑥, 𝑦⟩}
10	9	gen2 1795	. . . . . 6 ⊢ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}
11		19.29r2 1874	. . . . . . 7 ⊢ ((∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}) → ∃𝑥∃𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun ◡{⟨𝑥, 𝑦⟩}))
12		cnveq 5864	. . . . . . . . . 10 ⊢ ({𝑝} = {⟨𝑥, 𝑦⟩} → ◡{𝑝} = ◡{⟨𝑥, 𝑦⟩})
13	12	funeqd 6568	. . . . . . . . 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 1931	. . . . . . 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 586	. . . . 5 ⊢ (Fun {𝑝} → Fun ◡{𝑝})
18	17	ax-gen 1794	. . . 4 ⊢ ∀𝑝(Fun {𝑝} → Fun ◡{𝑝})
19		19.29r 1873	. . . . 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 5864	. . . . . . . . 9 ⊢ (𝐹 = {𝑝} → ◡𝐹 = ◡{𝑝})
22	21	funeqd 6568	. . . . . . . 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 1929	. . . . 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 691	. . 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 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {csn 4606  ⟨cop 4612   class class class wbr 5123  ^◡ccnv 5664  Rel wrel 5670  Fun wfun 6535  1_oc1o 8481   ≈ cen 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1o 8488  df-en 8968

This theorem is referenced by:  spthcycl  35109