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Theorem cardval2 8855
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 9406. This theorem could be used to give us a simpler definition of card in place of df-card 8803. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval2
StepHypRef Expression
1 cardsdomel 8838 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
21ancoms 468 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
32pm5.32da 674 . . . 4 (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))))
4 cardon 8808 . . . . . 6 (card‘𝐴) ∈ On
54oneli 5873 . . . . 5 (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
65pm4.71ri 666 . . . 4 (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
73, 6syl6rbbr 279 . . 3 (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥𝐴)))
87abbi2dv 2771 . 2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)})
9 df-rab 2950 . 2 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
108, 9syl6eqr 2703 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  {crab 2945   class class class wbr 4685  dom cdm 5143  Oncon0 5761  cfv 5926  csdm 7996  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803
This theorem is referenced by:  ondomon  9423  alephsuc3  9440
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