MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardidm Structured version   Visualization version   GIF version

Theorem cardidm 9461
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 9455 . . . . . . . 8 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 8606 . . . . . . 7 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 entr 8607 . . . . . . . 8 ((𝑦𝐴𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴))
43expcom 417 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
52, 4syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
6 entr 8607 . . . . . . . 8 ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦𝐴)
76expcom 417 . . . . . . 7 ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
81, 7syl 17 . . . . . 6 (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦𝐴))
95, 8impbid 215 . . . . 5 (𝐴 ∈ dom card → (𝑦𝐴𝑦 ≈ (card‘𝐴)))
109rabbidv 3381 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
1110inteqd 4841 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12 cardval3 9454 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
13 cardon 9446 . . . 4 (card‘𝐴) ∈ On
14 oncardval 9457 . . . 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 2784 . 2 (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17 card0 9460 . . 3 (card‘∅) = ∅
18 ndmfv 6704 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1918fveq2d 6678 . . 3 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
2017, 19, 183eqtr4a 2799 . 2 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
2116, 20pm2.61i 185 1 (card‘(card‘𝐴)) = (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2114  {crab 3057  c0 4211   cint 4836   class class class wbr 5030  dom cdm 5525  Oncon0 6172  cfv 6339  cen 8552  cardccrd 9437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-er 8320  df-en 8556  df-card 9441
This theorem is referenced by:  oncard  9462  cardlim  9474  cardiun  9484  alephnbtwn2  9572  infenaleph  9591  dfac12k  9647  pwsdompw  9704  cardcf  9752  cfeq0  9756  cfflb  9759  alephval2  10072  cfpwsdom  10084  gch2  10175  tskcard  10281  hashcard  13808  iscard4  40694
  Copyright terms: Public domain W3C validator