Theorem cardval3 9971

Description: An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
cardval3	⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3485	. 2 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V)
2		isnum2 9964	. . . 4 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
3		rabn0 4369	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		intex 5319	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
5	2, 3, 4	3bitr2i 299	. . 3 ⊢ (𝐴 ∈ dom card ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
6	5	biimpi 216	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
7		breq2 5128	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rabbidv 3428	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
9	8	inteqd 4932	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
10		df-card 9958	. . 3 ⊢ card = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦})
11	9, 10	fvmptg 6989	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
12	1, 6, 11	syl2anc 584	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  {crab 3420  Vcvv 3464  ∅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-pr 5407

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-fv 6544  df-en 8965  df-card 9958

This theorem is referenced by:  cardid2  9972  oncardval  9974  cardidm  9978  cardne  9984  cardval  10565