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Theorem cardval3 9949
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 3487 . 2 (𝐴 ∈ dom card β†’ 𝐴 ∈ V)
2 isnum2 9942 . . . 4 (𝐴 ∈ dom card ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
3 rabn0 4380 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
4 intex 5330 . . . 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 5145 . . . . 5 (𝑦 = 𝐴 β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ β‰ˆ 𝐴))
87rabbidv 3434 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
98inteqd 4948 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦} = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
10 df-card 9936 . . 3 card = (𝑦 ∈ V ↦ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝑦})
119, 10fvmptg 6990 . 2 ((𝐴 ∈ V ∧ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} ∈ V) β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
121, 6, 11syl2anc 583 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 2934  βˆƒwrex 3064  {crab 3426  Vcvv 3468  βˆ…c0 4317  βˆ© cint 4943   class class class wbr 5141  dom cdm 5669  Oncon0 6358  β€˜cfv 6537   β‰ˆ cen 8938  cardccrd 9932
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-en 8942  df-card 9936
This theorem is referenced by:  cardid2  9950  oncardval  9952  cardidm  9956  cardne  9962  cardval  10543
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