Theorem funen1cnv 32960

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 8765	. . 3 ⊢ (𝐹 ≈ 1o ↔ ∃𝑝 𝐹 = {𝑝})
2		funrel 6435	. . . . . . . 8 ⊢ (Fun {𝑝} → Rel {𝑝})
3		vsnid 4595	. . . . . . . 8 ⊢ 𝑝 ∈ {𝑝}
4		elrel 5697	. . . . . . . 8 ⊢ ((Rel {𝑝} ∧ 𝑝 ∈ {𝑝}) → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
5	2, 3, 4	sylancl 585	. . . . . . 7 ⊢ (Fun {𝑝} → ∃𝑥∃𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
6		sneq 4568	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → {𝑝} = {⟨𝑥, 𝑦⟩})
7	6	2eximi 1839	. . . . . . 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 6468	. . . . . . 7 ⊢ Fun ◡{⟨𝑥, 𝑦⟩}
10	9	gen2 1800	. . . . . 6 ⊢ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}
11		19.29r2 1879	. . . . . . 7 ⊢ ((∃𝑥∃𝑦{𝑝} = {⟨𝑥, 𝑦⟩} ∧ ∀𝑥∀𝑦Fun ◡{⟨𝑥, 𝑦⟩}) → ∃𝑥∃𝑦({𝑝} = {⟨𝑥, 𝑦⟩} ∧ Fun ◡{⟨𝑥, 𝑦⟩}))
12		cnveq 5771	. . . . . . . . . 10 ⊢ ({𝑝} = {⟨𝑥, 𝑦⟩} → ◡{𝑝} = ◡{⟨𝑥, 𝑦⟩})
13	12	funeqd 6440	. . . . . . . . 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 1936	. . . . . . 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 585	. . . . 5 ⊢ (Fun {𝑝} → Fun ◡{𝑝})
18	17	ax-gen 1799	. . . 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 6438	. . . . . . . 8 ⊢ (𝐹 = {𝑝} → (Fun 𝐹 ↔ Fun {𝑝}))
21		cnveq 5771	. . . . . . . . 9 ⊢ (𝐹 = {𝑝} → ◡𝐹 = ◡{𝑝})
22	21	funeqd 6440	. . . . . . . 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 1934	. . . . 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 687	. . 3 ⊢ (∃𝑝 𝐹 = {𝑝} → (Fun 𝐹 → Fun ◡𝐹))
28	1, 27	sylbi 216	. 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 1783   ∈ wcel 2108  {csn 4558  ⟨cop 4564   class class class wbr 5070  ^◡ccnv 5579  Rel wrel 5585  Fun wfun 6412  1_oc1o 8260   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-en 8692

This theorem is referenced by:  spthcycl  32991