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Theorem cardval3 9895
Description: An alternate definition of the value of (cardβ€˜π΄) 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 β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardval3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3466 . 2 (𝐴 ∈ dom card β†’ 𝐴 ∈ V)
2 isnum2 9888 . . . 4 (𝐴 ∈ dom card ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
3 rabn0 4350 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
4 intex 5299 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} β‰  βˆ… ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V)
52, 3, 43bitr2i 299 . . 3 (𝐴 ∈ dom card ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V)
65biimpi 215 . 2 (𝐴 ∈ dom card β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V)
7 breq2 5114 . . . . 5 (𝑦 = 𝐴 β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ β‰ˆ 𝐴))
87rabbidv 3418 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
98inteqd 4917 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
10 df-card 9882 . . 3 card = (𝑦 ∈ V ↦ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦})
119, 10fvmptg 6951 . 2 ((𝐴 ∈ V ∧ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V) β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
121, 6, 11syl2anc 585 1 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  {crab 3410  Vcvv 3448  βˆ…c0 4287  βˆ© cint 4912   class class class wbr 5110  dom cdm 5638  Oncon0 6322  β€˜cfv 6501   β‰ˆ cen 8887  cardccrd 9878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  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 6325  df-on 6326  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-en 8891  df-card 9882
This theorem is referenced by:  cardid2  9896  oncardval  9898  cardidm  9902  cardne  9908  cardval  10489
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