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Theorem cardidm 9380
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 9374 . . . . . . . 8 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 8552 . . . . . . 7 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 entr 8553 . . . . . . . 8 ((𝑦𝐴𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴))
43expcom 416 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
52, 4syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
6 entr 8553 . . . . . . . 8 ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦𝐴)
76expcom 416 . . . . . . 7 ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
81, 7syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
95, 8impbid 214 . . . . 5 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
109rabbidv 3479 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1110inteqd 4872 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12 cardval3 9373 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
13 cardon 9365 . . . 4 (card‘𝐴) ∈ On
14 oncardval 9376 . . . 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 2865 . 2 (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17 card0 9379 . . 3 (card‘∅) = ∅
18 ndmfv 6693 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1918fveq2d 6667 . . 3 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
2017, 19, 183eqtr4a 2880 . 2 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
2116, 20pm2.61i 184 1 (card‘(card‘𝐴)) = (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1530  wcel 2107  {crab 3140  c0 4289   cint 4867   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cfv 6348  cen 8498  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8281  df-en 8502  df-card 9360
This theorem is referenced by:  oncard  9381  cardlim  9393  cardiun  9403  alephnbtwn2  9490  infenaleph  9509  dfac12k  9565  pwsdompw  9618  cardcf  9666  cfeq0  9670  cfflb  9673  alephval2  9986  cfpwsdom  9998  gch2  10089  tskcard  10195  hashcard  13708  iscard4  39885
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