Theorem cardval3 9905

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 3468	. 2 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V)
2		isnum2 9898	. . . 4 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
3		rabn0 4352	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		intex 5299	. . . 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 5111	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rabbidv 3413	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
9	8	inteqd 4915	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
10		df-card 9892	. . 3 ⊢ card = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦})
11	9, 10	fvmptg 6966	. 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 2925  ∃wrex 3053  {crab 3405  Vcvv 3447  ∅c0 4296  ∩ cint 4910   class class class wbr 5107  dom cdm 5638  Oncon0 6332  ‘cfv 6511   ≈ cen 8915  cardccrd 9888

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-en 8919  df-card 9892

This theorem is referenced by:  cardid2  9906  oncardval  9908  cardidm  9912  cardne  9918  cardval  10499