Theorem cnvfi 9176

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5362. (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 5871	. . 3 ⊢ (𝑥 = ∅ → ◡𝑥 = ◡∅)
2	1	eleq1d 2818	. 2 ⊢ (𝑥 = ∅ → (◡𝑥 ∈ Fin ↔ ◡∅ ∈ Fin))
3		cnveq 5871	. . 3 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
4	3	eleq1d 2818	. 2 ⊢ (𝑥 = 𝑦 → (◡𝑥 ∈ Fin ↔ ◡𝑦 ∈ Fin))
5		cnveq 5871	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ◡𝑥 = ◡(𝑦 ∪ {𝑧}))
6	5	eleq1d 2818	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (◡𝑥 ∈ Fin ↔ ◡(𝑦 ∪ {𝑧}) ∈ Fin))
7		cnveq 5871	. . 3 ⊢ (𝑥 = 𝐴 → ◡𝑥 = ◡𝐴)
8	7	eleq1d 2818	. 2 ⊢ (𝑥 = 𝐴 → (◡𝑥 ∈ Fin ↔ ◡𝐴 ∈ Fin))
9		cnv0 6137	. . 3 ⊢ ◡∅ = ∅
10		0fin 9167	. . 3 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2829	. 2 ⊢ ◡∅ ∈ Fin
12		cnvun 6139	. . . 4 ⊢ ◡(𝑦 ∪ {𝑧}) = (◡𝑦 ∪ ◡{𝑧})
13		elvv 5748	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14		sneq 4637	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15		cnveq 5871	. . . . . . . . . . 11 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = ◡{⟨𝑢, 𝑣⟩})
16		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
17		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
18	16, 17	cnvsn 6222	. . . . . . . . . . 11 ⊢ ◡{⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
19	15, 18	eqtrdi 2788	. . . . . . . . . 10 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
20	14, 19	syl 17	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
21		snfi 9040	. . . . . . . . 9 ⊢ {⟨𝑣, 𝑢⟩} ∈ Fin
22	20, 21	eqeltrdi 2841	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
23	22	exlimivv 1935	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
24	13, 23	sylbi 216	. . . . . 6 ⊢ (𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
25		dfdm4 5893	. . . . . . . . 9 ⊢ dom {𝑧} = ran ◡{𝑧}
26		dmsnn0 6203	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
27	26	biimpri 227	. . . . . . . . . 10 ⊢ (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
28	27	necon1bi 2969	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
29	25, 28	eqtr3id 2786	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ (V × V) → ran ◡{𝑧} = ∅)
30		relcnv 6100	. . . . . . . . 9 ⊢ Rel ◡{𝑧}
31		relrn0 5966	. . . . . . . . 9 ⊢ (Rel ◡{𝑧} → (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅)
33	29, 32	sylibr 233	. . . . . . 7 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} = ∅)
34	33, 10	eqeltrdi 2841	. . . . . 6 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
35	24, 34	pm2.61i 182	. . . . 5 ⊢ ◡{𝑧} ∈ Fin
36		unfi 9168	. . . . 5 ⊢ ((◡𝑦 ∈ Fin ∧ ◡{𝑧} ∈ Fin) → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
37	35, 36	mpan2 689	. . . 4 ⊢ (◡𝑦 ∈ Fin → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
38	12, 37	eqeltrid 2837	. . 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 9160	1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474   ∪ cun 3945  ∅c0 4321  {csn 4627  ⟨cop 4633   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  ran crn 5676  Rel wrel 5680  Fincfn 8935

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-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-fin 8939

This theorem is referenced by:  f1oenfirn  9179  f1domfi  9180  sbthfilem  9197  rnfi  9331  fsumcnv  15715  fprodcnv  15923  gsumcom3  19840  gsummpt2co  32187  gsumhashmul  32195