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Theorem cardidm 9377
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 9371 . . . . . . . 8 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 8549 . . . . . . 7 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 entr 8550 . . . . . . . 8 ((𝑦𝐴𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴))
43expcom 414 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
52, 4syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
6 entr 8550 . . . . . . . 8 ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦𝐴)
76expcom 414 . . . . . . 7 ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
81, 7syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
95, 8impbid 213 . . . . 5 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
109rabbidv 3481 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1110inteqd 4874 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12 cardval3 9370 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
13 cardon 9362 . . . 4 (card‘𝐴) ∈ On
14 oncardval 9373 . . . 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 2867 . 2 (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17 card0 9376 . . 3 (card‘∅) = ∅
18 ndmfv 6694 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1918fveq2d 6668 . . 3 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
2017, 19, 183eqtr4a 2882 . 2 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
2116, 20pm2.61i 183 1 (card‘(card‘𝐴)) = (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  {crab 3142  c0 4290   cint 4869   class class class wbr 5058  dom cdm 5549  Oncon0 6185  cfv 6349  cen 8495  cardccrd 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-er 8279  df-en 8499  df-card 9357
This theorem is referenced by:  oncard  9378  cardlim  9390  cardiun  9400  alephnbtwn2  9487  infenaleph  9506  dfac12k  9562  pwsdompw  9615  cardcf  9663  cfeq0  9667  cfflb  9670  alephval2  9983  cfpwsdom  9995  gch2  10086  tskcard  10192  hashcard  13706  iscard4  39780
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