Theorem cnvfi 9243

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5383. (Revised by BTernaryTau, 9-Sep-2024.)

Assertion

Ref	Expression
cnvfi	⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Proof of Theorem cnvfi

Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnveq 5898	. . 3 ⊢ (𝑥 = ∅ → ◡𝑥 = ◡∅)
2	1	eleq1d 2829	. 2 ⊢ (𝑥 = ∅ → (◡𝑥 ∈ Fin ↔ ◡∅ ∈ Fin))
3		cnveq 5898	. . 3 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
4	3	eleq1d 2829	. 2 ⊢ (𝑥 = 𝑦 → (◡𝑥 ∈ Fin ↔ ◡𝑦 ∈ Fin))
5		cnveq 5898	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ◡𝑥 = ◡(𝑦 ∪ {𝑧}))
6	5	eleq1d 2829	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (◡𝑥 ∈ Fin ↔ ◡(𝑦 ∪ {𝑧}) ∈ Fin))
7		cnveq 5898	. . 3 ⊢ (𝑥 = 𝐴 → ◡𝑥 = ◡𝐴)
8	7	eleq1d 2829	. 2 ⊢ (𝑥 = 𝐴 → (◡𝑥 ∈ Fin ↔ ◡𝐴 ∈ Fin))
9		cnv0 6172	. . 3 ⊢ ◡∅ = ∅
10		0fi 9108	. . 3 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2840	. 2 ⊢ ◡∅ ∈ Fin
12		cnvun 6174	. . . 4 ⊢ ◡(𝑦 ∪ {𝑧}) = (◡𝑦 ∪ ◡{𝑧})
13		elvv 5774	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14		sneq 4658	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15		cnveq 5898	. . . . . . . . . . 11 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = ◡{⟨𝑢, 𝑣⟩})
16		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
17		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
18	16, 17	cnvsn 6257	. . . . . . . . . . 11 ⊢ ◡{⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
19	15, 18	eqtrdi 2796	. . . . . . . . . 10 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
20	14, 19	syl 17	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
21		snfi 9109	. . . . . . . . 9 ⊢ {⟨𝑣, 𝑢⟩} ∈ Fin
22	20, 21	eqeltrdi 2852	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
23	22	exlimivv 1931	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
24	13, 23	sylbi 217	. . . . . 6 ⊢ (𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
25		dfdm4 5920	. . . . . . . . 9 ⊢ dom {𝑧} = ran ◡{𝑧}
26		dmsnn0 6238	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
27	26	biimpri 228	. . . . . . . . . 10 ⊢ (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
28	27	necon1bi 2975	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
29	25, 28	eqtr3id 2794	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ (V × V) → ran ◡{𝑧} = ∅)
30		relcnv 6134	. . . . . . . . 9 ⊢ Rel ◡{𝑧}
31		relrn0 5995	. . . . . . . . 9 ⊢ (Rel ◡{𝑧} → (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅)
33	29, 32	sylibr 234	. . . . . . 7 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} = ∅)
34	33, 10	eqeltrdi 2852	. . . . . 6 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
35	24, 34	pm2.61i 182	. . . . 5 ⊢ ◡{𝑧} ∈ Fin
36		unfi 9238	. . . . 5 ⊢ ((◡𝑦 ∈ Fin ∧ ◡{𝑧} ∈ Fin) → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
37	35, 36	mpan2 690	. . . 4 ⊢ (◡𝑦 ∈ Fin → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
38	12, 37	eqeltrid 2848	. . 3 ⊢ (◡𝑦 ∈ Fin → ◡(𝑦 ∪ {𝑧}) ∈ Fin)
39	38	a1i 11	. 2 ⊢ (𝑦 ∈ Fin → (◡𝑦 ∈ Fin → ◡(𝑦 ∪ {𝑧}) ∈ Fin))
40	2, 4, 6, 8, 11, 39	findcard2 9230	1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∪ cun 3974  ∅c0 4352  {csn 4648  ⟨cop 4654   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701  Rel wrel 5705  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-fin 9007

This theorem is referenced by:  f1oenfirn  9246  f1domfi  9247  sbthfilem  9264  fodomfir  9396  rnfi  9408  fsumcnv  15821  fprodcnv  16031  gsumcom3  20020  gsummpt2co  33031  gsumhashmul  33040