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Theorem cardval3 9983
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 3492 . 2 (𝐴 ∈ dom card β†’ 𝐴 ∈ V)
2 isnum2 9976 . . . 4 (𝐴 ∈ dom card ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
3 rabn0 4389 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
4 intex 5343 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} β‰  βˆ… ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V)
52, 3, 43bitr2i 298 . . 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 5156 . . . . 5 (𝑦 = 𝐴 β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ β‰ˆ 𝐴))
87rabbidv 3438 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
98inteqd 4958 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
10 df-card 9970 . . 3 card = (𝑦 ∈ V ↦ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦})
119, 10fvmptg 7008 . 2 ((𝐴 ∈ V ∧ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V) β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
121, 6, 11syl2anc 582 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 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067  {crab 3430  Vcvv 3473  βˆ…c0 4326  βˆ© cint 4953   class class class wbr 5152  dom cdm 5682  Oncon0 6374  β€˜cfv 6553   β‰ˆ cen 8967  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-en 8971  df-card 9970
This theorem is referenced by:  cardid2  9984  oncardval  9986  cardidm  9990  cardne  9996  cardval  10577
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