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Theorem cardidm 9902
Description: The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardidm (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄)

Proof of Theorem cardidm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardid2 9896 . . . . . . . 8 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21ensymd 8952 . . . . . . 7 (𝐴 ∈ dom card β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
3 entr 8953 . . . . . . . 8 ((𝑦 β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ (cardβ€˜π΄)) β†’ 𝑦 β‰ˆ (cardβ€˜π΄))
43expcom 415 . . . . . . 7 (𝐴 β‰ˆ (cardβ€˜π΄) β†’ (𝑦 β‰ˆ 𝐴 β†’ 𝑦 β‰ˆ (cardβ€˜π΄)))
52, 4syl 17 . . . . . 6 (𝐴 ∈ dom card β†’ (𝑦 β‰ˆ 𝐴 β†’ 𝑦 β‰ˆ (cardβ€˜π΄)))
6 entr 8953 . . . . . . . 8 ((𝑦 β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐴) β†’ 𝑦 β‰ˆ 𝐴)
76expcom 415 . . . . . . 7 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ (𝑦 β‰ˆ (cardβ€˜π΄) β†’ 𝑦 β‰ˆ 𝐴))
81, 7syl 17 . . . . . 6 (𝐴 ∈ dom card β†’ (𝑦 β‰ˆ (cardβ€˜π΄) β†’ 𝑦 β‰ˆ 𝐴))
95, 8impbid 211 . . . . 5 (𝐴 ∈ dom card β†’ (𝑦 β‰ˆ 𝐴 ↔ 𝑦 β‰ˆ (cardβ€˜π΄)))
109rabbidv 3418 . . . 4 (𝐴 ∈ dom card β†’ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} = {𝑦 ∈ On ∣ 𝑦 β‰ˆ (cardβ€˜π΄)})
1110inteqd 4917 . . 3 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ (cardβ€˜π΄)})
12 cardval3 9895 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
13 cardon 9887 . . . 4 (cardβ€˜π΄) ∈ On
14 oncardval 9898 . . . 4 ((cardβ€˜π΄) ∈ On β†’ (cardβ€˜(cardβ€˜π΄)) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ (cardβ€˜π΄)})
1513, 14mp1i 13 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜(cardβ€˜π΄)) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ (cardβ€˜π΄)})
1611, 12, 153eqtr4rd 2788 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄))
17 card0 9901 . . 3 (cardβ€˜βˆ…) = βˆ…
18 ndmfv 6882 . . . 4 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
1918fveq2d 6851 . . 3 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜βˆ…))
2017, 19, 183eqtr4a 2803 . 2 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄))
2116, 20pm2.61i 182 1 (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3410  βˆ…c0 4287  βˆ© cint 4912   class class class wbr 5110  dom cdm 5638  Oncon0 6322  β€˜cfv 6501   β‰ˆ cen 8887  cardccrd 9878
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-er 8655  df-en 8891  df-card 9882
This theorem is referenced by:  oncard  9903  cardlim  9915  cardiun  9925  alephnbtwn2  10015  infenaleph  10034  dfac12k  10090  pwsdompw  10147  cardcf  10195  cfeq0  10199  cfflb  10202  alephval2  10515  cfpwsdom  10527  gch2  10618  tskcard  10724  hashcard  14262  iscard4  41879
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