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Theorem cnvfi 9127
Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5321. (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 5830 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
21eleq1d 2819 . 2 (𝑥 = ∅ → (𝑥 ∈ Fin ↔ ∅ ∈ Fin))
3 cnveq 5830 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43eleq1d 2819 . 2 (𝑥 = 𝑦 → (𝑥 ∈ Fin ↔ 𝑦 ∈ Fin))
5 cnveq 5830 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
65eleq1d 2819 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ Fin ↔ (𝑦 ∪ {𝑧}) ∈ Fin))
7 cnveq 5830 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87eleq1d 2819 . 2 (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
9 cnv0 6094 . . 3 ∅ = ∅
10 0fin 9118 . . 3 ∅ ∈ Fin
119, 10eqeltri 2830 . 2 ∅ ∈ Fin
12 cnvun 6096 . . . 4 (𝑦 ∪ {𝑧}) = (𝑦{𝑧})
13 elvv 5707 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
14 sneq 4597 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
15 cnveq 5830 . . . . . . . . . . 11 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑢, 𝑣⟩})
16 vex 3448 . . . . . . . . . . . 12 𝑢 ∈ V
17 vex 3448 . . . . . . . . . . . 12 𝑣 ∈ V
1816, 17cnvsn 6179 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩} = {⟨𝑣, 𝑢⟩}
1915, 18eqtrdi 2789 . . . . . . . . . 10 ({𝑧} = {⟨𝑢, 𝑣⟩} → {𝑧} = {⟨𝑣, 𝑢⟩})
2014, 19syl 17 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑣, 𝑢⟩})
21 snfi 8991 . . . . . . . . 9 {⟨𝑣, 𝑢⟩} ∈ Fin
2220, 21eqeltrdi 2842 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2322exlimivv 1936 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} ∈ Fin)
2413, 23sylbi 216 . . . . . 6 (𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
25 dfdm4 5852 . . . . . . . . 9 dom {𝑧} = ran {𝑧}
26 dmsnn0 6160 . . . . . . . . . . 11 (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
2726biimpri 227 . . . . . . . . . 10 (dom {𝑧} ≠ ∅ → 𝑧 ∈ (V × V))
2827necon1bi 2969 . . . . . . . . 9 𝑧 ∈ (V × V) → dom {𝑧} = ∅)
2925, 28eqtr3id 2787 . . . . . . . 8 𝑧 ∈ (V × V) → ran {𝑧} = ∅)
30 relcnv 6057 . . . . . . . . 9 Rel {𝑧}
31 relrn0 5925 . . . . . . . . 9 (Rel {𝑧} → ({𝑧} = ∅ ↔ ran {𝑧} = ∅))
3230, 31ax-mp 5 . . . . . . . 8 ({𝑧} = ∅ ↔ ran {𝑧} = ∅)
3329, 32sylibr 233 . . . . . . 7 𝑧 ∈ (V × V) → {𝑧} = ∅)
3433, 10eqeltrdi 2842 . . . . . 6 𝑧 ∈ (V × V) → {𝑧} ∈ Fin)
3524, 34pm2.61i 182 . . . . 5 {𝑧} ∈ Fin
36 unfi 9119 . . . . 5 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦{𝑧}) ∈ Fin)
3735, 36mpan2 690 . . . 4 (𝑦 ∈ Fin → (𝑦{𝑧}) ∈ Fin)
3812, 37eqeltrid 2838 . . 3 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
3938a1i 11 . 2 (𝑦 ∈ Fin → (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin))
402, 4, 6, 8, 11, 39findcard2 9111 1 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wex 1782  wcel 2107  wne 2940  Vcvv 3444  cun 3909  c0 4283  {csn 4587  cop 4593   × cxp 5632  ccnv 5633  dom cdm 5634  ran crn 5635  Rel wrel 5639  Fincfn 8886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-en 8887  df-fin 8890
This theorem is referenced by:  f1oenfirn  9130  f1domfi  9131  sbthfilem  9148  rnfi  9282  fsumcnv  15663  fprodcnv  15871  gsumcom3  19760  gsummpt2co  31939  gsumhashmul  31947
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