Theorem cardidm 9978

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.

Step	Hyp	Ref	Expression
1		cardid2 9972	. . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 9024	. . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		entr 9025	. . . . . . . 8 ⊢ ((𝑦 ≈ 𝐴 ∧ 𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴))
4	3	expcom 413	. . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → (𝑦 ≈ 𝐴 → 𝑦 ≈ (card‘𝐴)))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 → 𝑦 ≈ (card‘𝐴)))
6		entr 9025	. . . . . . . 8 ⊢ ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦 ≈ 𝐴)
7	6	expcom 413	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦 ≈ 𝐴))
8	1, 7	syl 17	. . . . . 6 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦 ≈ 𝐴))
9	5, 8	impbid 212	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ (card‘𝐴)))
10	9	rabbidv 3428	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
11	10	inteqd 4932	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12		cardval3 9971	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
13		cardon 9963	. . . 4 ⊢ (card‘𝐴) ∈ On
14		oncardval 9974	. . . 4 ⊢ ((card‘𝐴) ∈ On → (card‘(card‘𝐴)) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
15	13, 14	mp1i 13	. . 3 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
16	11, 12, 15	3eqtr4rd 2782	. 2 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17		card0 9977	. . 3 ⊢ (card‘∅) = ∅
18		ndmfv 6916	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
19	18	fveq2d 6885	. . 3 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
20	17, 19, 18	3eqtr4a 2797	. 2 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
21	16, 20	pm2.61i 182	1 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3420  ∅c0 4313  ∩ cint 4927   class class class wbr 5124  dom cdm 5659  Oncon0 6357  ‘cfv 6536   ≈ cen 8961  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8724  df-en 8965  df-card 9958

This theorem is referenced by:  oncard  9979  cardlim  9991  cardiun  10001  alephnbtwn2  10091  infenaleph  10110  dfac12k  10167  pwsdompw  10222  cardcf  10271  cfeq0  10275  cfflb  10278  alephval2  10591  cfpwsdom  10603  gch2  10694  tskcard  10800  hashcard  14378  iscard4  43532