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Theorem cardval3 9953
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 ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐴 ∈ dom card → 𝐴 ∈ V)
2 isnum2 9946 . . . 4 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
3 rabn0 4385 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 intex 5337 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
52, 3, 43bitr2i 299 . . 3 (𝐴 ∈ dom card ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
65biimpi 215 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
7 breq2 5152 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rabbidv 3439 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
98inteqd 4955 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
10 df-card 9940 . . 3 card = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑥𝑦})
119, 10fvmptg 6996 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V) → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
121, 6, 11syl2anc 583 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wne 2939  wrex 3069  {crab 3431  Vcvv 3473  c0 4322   cint 4950   class class class wbr 5148  dom cdm 5676  Oncon0 6364  cfv 6543  cen 8942  cardccrd 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-en 8946  df-card 9940
This theorem is referenced by:  cardid2  9954  oncardval  9956  cardidm  9960  cardne  9966  cardval  10547
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