Theorem cardidm 9878

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 9872	. . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 8947	. . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		entr 8948	. . . . . . . 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 8948	. . . . . . . 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 3397	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
11	10	inteqd 4895	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12		cardval3 9871	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
13		cardon 9863	. . . 4 ⊢ (card‘𝐴) ∈ On
14		oncardval 9874	. . . 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 2783	. 2 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴))
17		card0 9877	. . 3 ⊢ (card‘∅) = ∅
18		ndmfv 6868	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
19	18	fveq2d 6840	. . 3 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅))
20	17, 19, 18	3eqtr4a 2798	. 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 1542   ∈ wcel 2114  {crab 3390  ∅c0 4274  ∩ cint 4890   class class class wbr 5086  dom cdm 5626  Oncon0 6319  ‘cfv 6494   ≈ cen 8885  cardccrd 9854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-er 8638  df-en 8889  df-card 9858

This theorem is referenced by:  oncard  9879  cardlim  9891  cardiun  9901  alephnbtwn2  9989  infenaleph  10008  dfac12k  10065  pwsdompw  10120  cardcf  10169  cfeq0  10173  cfflb  10176  alephval2  10490  cfpwsdom  10502  gch2  10593  tskcard  10699  hashcard  14312  iscard4  43984