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Theorem cnvfi 9214
Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5370. (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 5886 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
21eleq1d 2823 . 2 (𝑥 = ∅ → (𝑥 ∈ Fin ↔ ∅ ∈ Fin))
3 cnveq 5886 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43eleq1d 2823 . 2 (𝑥 = 𝑦 → (𝑥 ∈ Fin ↔ 𝑦 ∈ Fin))
5 cnveq 5886 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
65eleq1d 2823 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ Fin ↔ (𝑦 ∪ {𝑧}) ∈ Fin))
7 cnveq 5886 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87eleq1d 2823 . 2 (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
9 cnv0 6162 . . 3 ∅ = ∅
10 0fi 9080 . . 3 ∅ ∈ Fin
119, 10eqeltri 2834 . 2 ∅ ∈ Fin
12 cnvun 6164 . . . 4 (𝑦 ∪ {𝑧}) = (𝑦{𝑧})
13 elvv 5762 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14 sneq 4640 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15 cnveq 5886 . . . . . . . . . . 11 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑢, 𝑣⟩})
16 vex 3481 . . . . . . . . . . . 12 𝑢 ∈ V
17 vex 3481 . . . . . . . . . . . 12 𝑣 ∈ V
1816, 17cnvsn 6247 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
1915, 18eqtrdi 2790 . . . . . . . . . 10 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑣, 𝑢⟩})
2014, 19syl 17 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑣, 𝑢⟩})
21 snfi 9081 . . . . . . . . 9 {⟨𝑣, 𝑢⟩} ∈ Fin
2220, 21eqeltrdi 2846 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2322exlimivv 1929 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2413, 23sylbi 217 . . . . . 6 (𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
25 dfdm4 5908 . . . . . . . . 9 dom {𝑧} = ran {𝑧}
26 dmsnn0 6228 . . . . . . . . . . 11 (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
2726biimpri 228 . . . . . . . . . 10 (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
2827necon1bi 2966 . . . . . . . . 9 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
2925, 28eqtr3id 2788 . . . . . . . 8 𝑧 ∈ (V × V) → ran {𝑧} = ∅)
30 relcnv 6124 . . . . . . . . 9 Rel {𝑧}
31 relrn0 5985 . . . . . . . . 9 (Rel {𝑧} → ({𝑧} = ∅ ↔ ran {𝑧} = ∅))
3230, 31ax-mp 5 . . . . . . . 8 ({𝑧} = ∅ ↔ ran {𝑧} = ∅)
3329, 32sylibr 234 . . . . . . 7 𝑧 ∈ (V × V) → {𝑧} = ∅)
3433, 10eqeltrdi 2846 . . . . . 6 𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
3524, 34pm2.61i 182 . . . . 5 {𝑧} ∈ Fin
36 unfi 9209 . . . . 5 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦{𝑧}) ∈ Fin)
3735, 36mpan2 691 . . . 4 (𝑦 ∈ Fin → (𝑦{𝑧}) ∈ Fin)
3812, 37eqeltrid 2842 . . 3 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
3938a1i 11 . 2 (𝑦 ∈ Fin → (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin))
402, 4, 6, 8, 11, 39findcard2 9202 1 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1536  wex 1775  wcel 2105  wne 2937  Vcvv 3477  cun 3960  c0 4338  {csn 4630  cop 4636   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  Rel wrel 5693  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-en 8984  df-fin 8987
This theorem is referenced by:  f1oenfirn  9217  f1domfi  9218  sbthfilem  9235  fodomfir  9365  rnfi  9377  fsumcnv  15805  fprodcnv  16015  gsumcom3  20010  gsummpt2co  33033  gsumhashmul  33046
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