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Theorem cnvfi 9216
Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5365. (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.
StepHypRef Expression
1 cnveq 5884 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
21eleq1d 2826 . 2 (𝑥 = ∅ → (𝑥 ∈ Fin ↔ ∅ ∈ Fin))
3 cnveq 5884 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43eleq1d 2826 . 2 (𝑥 = 𝑦 → (𝑥 ∈ Fin ↔ 𝑦 ∈ Fin))
5 cnveq 5884 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
65eleq1d 2826 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ Fin ↔ (𝑦 ∪ {𝑧}) ∈ Fin))
7 cnveq 5884 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87eleq1d 2826 . 2 (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
9 cnv0 6160 . . 3 ∅ = ∅
10 0fi 9082 . . 3 ∅ ∈ Fin
119, 10eqeltri 2837 . 2 ∅ ∈ Fin
12 cnvun 6162 . . . 4 (𝑦 ∪ {𝑧}) = (𝑦{𝑧})
13 elvv 5760 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14 sneq 4636 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15 cnveq 5884 . . . . . . . . . . 11 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑢, 𝑣⟩})
16 vex 3484 . . . . . . . . . . . 12 𝑢 ∈ V
17 vex 3484 . . . . . . . . . . . 12 𝑣 ∈ V
1816, 17cnvsn 6246 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
1915, 18eqtrdi 2793 . . . . . . . . . 10 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑣, 𝑢⟩})
2014, 19syl 17 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑣, 𝑢⟩})
21 snfi 9083 . . . . . . . . 9 {⟨𝑣, 𝑢⟩} ∈ Fin
2220, 21eqeltrdi 2849 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2322exlimivv 1932 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2413, 23sylbi 217 . . . . . 6 (𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
25 dfdm4 5906 . . . . . . . . 9 dom {𝑧} = ran {𝑧}
26 dmsnn0 6227 . . . . . . . . . . 11 (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
2726biimpri 228 . . . . . . . . . 10 (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
2827necon1bi 2969 . . . . . . . . 9 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
2925, 28eqtr3id 2791 . . . . . . . 8 𝑧 ∈ (V × V) → ran {𝑧} = ∅)
30 relcnv 6122 . . . . . . . . 9 Rel {𝑧}
31 relrn0 5983 . . . . . . . . 9 (Rel {𝑧} → ({𝑧} = ∅ ↔ ran {𝑧} = ∅))
3230, 31ax-mp 5 . . . . . . . 8 ({𝑧} = ∅ ↔ ran {𝑧} = ∅)
3329, 32sylibr 234 . . . . . . 7 𝑧 ∈ (V × V) → {𝑧} = ∅)
3433, 10eqeltrdi 2849 . . . . . 6 𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
3524, 34pm2.61i 182 . . . . 5 {𝑧} ∈ Fin
36 unfi 9211 . . . . 5 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦{𝑧}) ∈ Fin)
3735, 36mpan2 691 . . . 4 (𝑦 ∈ Fin → (𝑦{𝑧}) ∈ Fin)
3812, 37eqeltrid 2845 . . 3 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
3938a1i 11 . 2 (𝑦 ∈ Fin → (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin))
402, 4, 6, 8, 11, 39findcard2 9204 1 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  cun 3949  c0 4333  {csn 4626  cop 4632   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  Rel wrel 5690  Fincfn 8985
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by:  f1oenfirn  9220  f1domfi  9221  sbthfilem  9238  fodomfir  9368  rnfi  9380  fsumcnv  15809  fprodcnv  16019  gsumcom3  19996  gsummpt2co  33051  gsumhashmul  33064
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