Theorem cnvfi 9105

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5311. (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 5823	. . 3 ⊢ (𝑥 = ∅ → ◡𝑥 = ◡∅)
2	1	eleq1d 2822	. 2 ⊢ (𝑥 = ∅ → (◡𝑥 ∈ Fin ↔ ◡∅ ∈ Fin))
3		cnveq 5823	. . 3 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
4	3	eleq1d 2822	. 2 ⊢ (𝑥 = 𝑦 → (◡𝑥 ∈ Fin ↔ ◡𝑦 ∈ Fin))
5		cnveq 5823	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ◡𝑥 = ◡(𝑦 ∪ {𝑧}))
6	5	eleq1d 2822	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (◡𝑥 ∈ Fin ↔ ◡(𝑦 ∪ {𝑧}) ∈ Fin))
7		cnveq 5823	. . 3 ⊢ (𝑥 = 𝐴 → ◡𝑥 = ◡𝐴)
8	7	eleq1d 2822	. 2 ⊢ (𝑥 = 𝐴 → (◡𝑥 ∈ Fin ↔ ◡𝐴 ∈ Fin))
9		cnv0 6098	. . 3 ⊢ ◡∅ = ∅
10		0fi 8984	. . 3 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2833	. 2 ⊢ ◡∅ ∈ Fin
12		cnvun 6101	. . . 4 ⊢ ◡(𝑦 ∪ {𝑧}) = (◡𝑦 ∪ ◡{𝑧})
13		elvv 5700	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14		sneq 4591	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15		cnveq 5823	. . . . . . . . . . 11 ⊢ ({𝑧} = {⟨𝑢, 𝑣⟩} → ◡{𝑧} = ◡{⟨𝑢, 𝑣⟩})
16		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
17		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
18	16, 17	cnvsn 6185	. . . . . . . . . . 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 8985	. . . . . . . . 9 ⊢ {⟨𝑣, 𝑢⟩} ∈ Fin
22	20, 21	eqeltrdi 2845	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
23	22	exlimivv 1934	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ◡{𝑧} ∈ Fin)
24	13, 23	sylbi 217	. . . . . 6 ⊢ (𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
25		dfdm4 5845	. . . . . . . . 9 ⊢ dom {𝑧} = ran ◡{𝑧}
26		dmsnn0 6166	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
27	26	biimpri 228	. . . . . . . . . 10 ⊢ (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
28	27	necon1bi 2961	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
29	25, 28	eqtr3id 2786	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ (V × V) → ran ◡{𝑧} = ∅)
30		relcnv 6064	. . . . . . . . 9 ⊢ Rel ◡{𝑧}
31		relrn0 5923	. . . . . . . . 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 2845	. . . . . 6 ⊢ (¬ 𝑧 ∈ (V × V) → ◡{𝑧} ∈ Fin)
35	24, 34	pm2.61i 182	. . . . 5 ⊢ ◡{𝑧} ∈ Fin
36		unfi 9100	. . . . 5 ⊢ ((◡𝑦 ∈ Fin ∧ ◡{𝑧} ∈ Fin) → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
37	35, 36	mpan2 692	. . . 4 ⊢ (◡𝑦 ∈ Fin → (◡𝑦 ∪ ◡{𝑧}) ∈ Fin)
38	12, 37	eqeltrid 2841	. . 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 9094	1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  Vcvv 3441   ∪ cun 3900  ∅c0 4286  {csn 4581  ⟨cop 4587   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  Rel wrel 5630  Fincfn 8888

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7812  df-1o 8400  df-en 8889  df-fin 8892

This theorem is referenced by:  f1oenfirn  9109  f1domfi  9110  sbthfilem  9127  fodomfir  9233  rnfi  9245  fsumcnv  15701  fprodcnv  15911  gsumcom3  19912  gsummpt2co  33134  gsumhashmul  33153