Theorem cnvfi 9090

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5304. (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 5816	. . 3 ⊢ (𝑥 = ∅ → ◡𝑥 = ◡∅)
2	1	eleq1d 2813	. 2 ⊢ (𝑥 = ∅ → (◡𝑥 ∈ Fin ↔ ◡∅ ∈ Fin))
3		cnveq 5816	. . 3 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
4	3	eleq1d 2813	. 2 ⊢ (𝑥 = 𝑦 → (◡𝑥 ∈ Fin ↔ ◡𝑦 ∈ Fin))
5		cnveq 5816	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ◡𝑥 = ◡(𝑦 ∪ {𝑧}))
6	5	eleq1d 2813	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (◡𝑥 ∈ Fin ↔ ◡(𝑦 ∪ {𝑧}) ∈ Fin))
7		cnveq 5816	. . 3 ⊢ (𝑥 = 𝐴 → ◡𝑥 = ◡𝐴)
8	7	eleq1d 2813	. 2 ⊢ (𝑥 = 𝐴 → (◡𝑥 ∈ Fin ↔ ◡𝐴 ∈ Fin))
9		cnv0 6089	. . 3 ⊢ ◡∅ = ∅
10		0fi 8967	. . 3 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2824	. 2 ⊢ ◡∅ ∈ Fin
12		cnvun 6091	. . . 4 ⊢ ◡(𝑦 ∪ {𝑧}) = (◡𝑦 ∪ ◡{𝑧})
13		elvv 5694	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14		sneq 4587	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15		cnveq 5816	. . . . . . . . . . 11 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = ◡{⟨𝑢, 𝑣⟩})
16		vex 3440	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
17		vex 3440	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
18	16, 17	cnvsn 6175	. . . . . . . . . . 11 ⊢ ◡{⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
19	15, 18	eqtrdi 2780	. . . . . . . . . 10 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
20	14, 19	syl 17	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
21		snfi 8968	. . . . . . . . 9 ⊢ {⟨𝑣, 𝑢⟩} ∈ Fin
22	20, 21	eqeltrdi 2836	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
23	22	exlimivv 1932	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
24	13, 23	sylbi 217	. . . . . 6 ⊢ (𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
25		dfdm4 5838	. . . . . . . . 9 ⊢ dom {𝑧} = ran ◡{𝑧}
26		dmsnn0 6156	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
27	26	biimpri 228	. . . . . . . . . 10 ⊢ (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
28	27	necon1bi 2953	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
29	25, 28	eqtr3id 2778	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ (V × V) → ran ◡{𝑧} = ∅)
30		relcnv 6055	. . . . . . . . 9 ⊢ Rel ◡{𝑧}
31		relrn0 5914	. . . . . . . . 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 2836	. . . . . 6 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
35	24, 34	pm2.61i 182	. . . . 5 ⊢ ◡{𝑧} ∈ Fin
36		unfi 9085	. . . . 5 ⊢ ((◡𝑦 ∈ Fin ∧ ◡{𝑧} ∈ Fin) → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
37	35, 36	mpan2 691	. . . 4 ⊢ (◡𝑦 ∈ Fin → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
38	12, 37	eqeltrid 2832	. . 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 9078	1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∪ cun 3901  ∅c0 4284  {csn 4577  ⟨cop 4583   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620  Rel wrel 5624  Fincfn 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-en 8873  df-fin 8876

This theorem is referenced by:  f1oenfirn  9094  f1domfi  9095  sbthfilem  9112  fodomfir  9218  rnfi  9230  fsumcnv  15680  fprodcnv  15890  gsumcom3  19857  gsummpt2co  33010  gsumhashmul  33023