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Theorem cardval3 9029
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 3365 . 2 (𝐴 ∈ dom card → 𝐴 ∈ V)
2 isnum2 9022 . . . 4 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
3 rabn0 4122 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 intex 4978 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
52, 3, 43bitr2i 290 . . 3 (𝐴 ∈ dom card ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
65biimpi 207 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
7 breq2 4813 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rabbidv 3338 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
98inteqd 4638 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
10 df-card 9016 . . 3 card = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑥𝑦})
119, 10fvmptg 6469 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V) → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
121, 6, 11syl2anc 579 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  wne 2937  wrex 3056  {crab 3059  Vcvv 3350  c0 4079   cint 4633   class class class wbr 4809  dom cdm 5277  Oncon0 5908  cfv 6068  cen 8157  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-en 8161  df-card 9016
This theorem is referenced by:  cardid2  9030  oncardval  9032  cardidm  9036  cardne  9042  cardval  9621
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