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Theorem cardidm 8996
Description: The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardidm (card‘(card‘𝐴)) = (card‘𝐴)

Proof of Theorem cardidm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardid2 8990 . . . . . . . 8 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 8175 . . . . . . 7 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 entr 8176 . . . . . . . 8 ((𝑦𝐴𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴))
43expcom 450 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
52, 4syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
6 entr 8176 . . . . . . . 8 ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦𝐴)
76expcom 450 . . . . . . 7 ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
81, 7syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
95, 8impbid 202 . . . . 5 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
109rabbidv 3330 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1110inteqd 4633 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12 cardval3 8989 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
13 cardon 8981 . . . 4 (card‘𝐴) ∈ On
14 oncardval 8992 . . . 4 ((card‘𝐴) ∈ On → (card‘(card‘𝐴)) = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1513, 14mp1i 13 . . 3 (𝐴 ∈ dom card → (card‘(card‘𝐴)) = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1611, 12, 153eqtr4rd 2806 . 2 (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17 card0 8995 . . 3 (card‘∅) = ∅
18 ndmfv 6381 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1918fveq2d 6358 . . 3 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
2017, 19, 183eqtr4a 2821 . 2 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
2116, 20pm2.61i 176 1 (card‘(card‘𝐴)) = (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wcel 2140  {crab 3055  c0 4059   cint 4628   class class class wbr 4805  dom cdm 5267  Oncon0 5885  cfv 6050  cen 8121  cardccrd 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ord 5888  df-on 5889  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-er 7914  df-en 8125  df-card 8976
This theorem is referenced by:  oncard  8997  cardlim  9009  cardiun  9019  alephnbtwn2  9106  infenaleph  9125  dfac12k  9182  pwsdompw  9239  cardcf  9287  cfeq0  9291  cfflb  9294  alephval2  9607  cfpwsdom  9619  gch2  9710  tskcard  9816  hashcard  13359
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