Theorem cardidm 9875

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 9869	. . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 8946	. . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		entr 8947	. . . . . . . 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 8947	. . . . . . . 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 3407	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
11	10	inteqd 4908	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)})
12		cardval3 9868	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
13		cardon 9860	. . . 4 ⊢ (card‘𝐴) ∈ On
14		oncardval 9871	. . . 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 9874	. . 3 ⊢ (card‘∅) = ∅
18		ndmfv 6867	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
19	18	fveq2d 6839	. . 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 3400  ∅c0 4286  ∩ cint 4903   class class class wbr 5099  dom cdm 5625  Oncon0 6318  ‘cfv 6493   ≈ cen 8884  cardccrd 9851

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 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-er 8637  df-en 8888  df-card 9855

This theorem is referenced by:  oncard  9876  cardlim  9888  cardiun  9898  alephnbtwn2  9986  infenaleph  10005  dfac12k  10062  pwsdompw  10117  cardcf  10166  cfeq0  10170  cfflb  10173  alephval2  10487  cfpwsdom  10499  gch2  10590  tskcard  10696  hashcard  14282  iscard4  43841