Theorem cnvfi 9146

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5324. (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 5847	. . 3 ⊢ (𝑥 = ∅ → ◡𝑥 = ◡∅)
2	1	eleq1d 2849	. 2 ⊢ (𝑥 = ∅ → (◡𝑥 ∈ Fin ↔ ◡∅ ∈ Fin))
3		cnveq 5847	. . 3 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
4	3	eleq1d 2849	. 2 ⊢ (𝑥 = 𝑦 → (◡𝑥 ∈ Fin ↔ ◡𝑦 ∈ Fin))
5		cnveq 5847	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ◡𝑥 = ◡(𝑦 ∪ {𝑧}))
6	5	eleq1d 2849	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (◡𝑥 ∈ Fin ↔ ◡(𝑦 ∪ {𝑧}) ∈ Fin))
7		cnveq 5847	. . 3 ⊢ (𝑥 = 𝐴 → ◡𝑥 = ◡𝐴)
8	7	eleq1d 2849	. 2 ⊢ (𝑥 = 𝐴 → (◡𝑥 ∈ Fin ↔ ◡𝐴 ∈ Fin))
9		cnv0 5857	. . 3 ⊢ ◡∅ = ∅
10		0fi 9025	. . 3 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2860	. 2 ⊢ ◡∅ ∈ Fin
12		cnvun 6128	. . . 4 ⊢ ◡(𝑦 ∪ {𝑧}) = (◡𝑦 ∪ ◡{𝑧})
13		elvv 5724	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14		sneq 4594	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15		cnveq 5847	. . . . . . . . . . 11 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = ◡{⟨𝑢, 𝑣⟩})
16		vex 3460	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
17		vex 3460	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
18	16, 17	cnvsn 6215	. . . . . . . . . . 11 ⊢ ◡{⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
19	15, 18	eqtrdi 2815	. . . . . . . . . 10 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
20	14, 19	syl 17	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} = {⟨𝑣, 𝑢⟩})
21		snfi 9026	. . . . . . . . 9 ⊢ {⟨𝑣, 𝑢⟩} ∈ Fin
22	20, 21	eqeltrdi 2872	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
23	22	exlimivv 1954	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
24	13, 23	sylbi 219	. . . . . 6 ⊢ (𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
25		dfdm4 5873	. . . . . . . . 9 ⊢ dom {𝑧} = ran ◡{𝑧}
26		dmsnn0 6196	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
27	26	biimpri 230	. . . . . . . . . 10 ⊢ (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
28	27	necon1bi 2987	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
29	25, 28	eqtr3id 2813	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ (V × V) → ran ◡{𝑧} = ∅)
30		relcnv 6095	. . . . . . . . 9 ⊢ Rel ◡{𝑧}
31		relrn0 5951	. . . . . . . . 9 ⊢ (Rel ◡{𝑧} → (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ (◡{𝑧} = ∅ ↔ ran ◡{𝑧} = ∅)
33	29, 32	sylibr 236	. . . . . . 7 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} = ∅)
34	33, 10	eqeltrdi 2872	. . . . . 6 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
35	24, 34	pm2.61i 183	. . . . 5 ⊢ ◡{𝑧} ∈ Fin
36		unfi 9141	. . . . 5 ⊢ ((◡𝑦 ∈ Fin ∧ ◡{𝑧} ∈ Fin) → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
37	35, 36	mpan2 701	. . . 4 ⊢ (◡𝑦 ∈ Fin → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
38	12, 37	eqeltrid 2868	. . 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 9135	1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  Vcvv 3456   ∪ cun 3904  ∅c0 4287  {csn 4584  ⟨cop 4590   × cxp 5647  ^◡ccnv 5648  dom cdm 5649  ran crn 5650  Rel wrel 5654  Fincfn 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-en 8930  df-fin 8933

This theorem is referenced by:  f1oenfirn  9150  f1domfi  9151  sbthfilem  9168  fodomfir  9274  rnfi  9285  fsumcnv  15802  fprodcnv  16015  gsumcom3  20020  gsummpt2co  33230  gsumhashmul  33249