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Theorem cardval3 8638
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 3184 . 2 (𝐴 ∈ dom card → 𝐴 ∈ V)
2 isnum2 8631 . . . 4 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
3 rabn0 3911 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 intex 4742 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
52, 3, 43bitr2i 286 . . 3 (𝐴 ∈ dom card ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
65biimpi 204 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
7 breq2 4581 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rabbidv 3163 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
98inteqd 4409 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
10 df-card 8625 . . 3 card = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑥𝑦})
119, 10fvmptg 6174 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V) → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
121, 6, 11syl2anc 690 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wne 2779  wrex 2896  {crab 2899  Vcvv 3172  c0 3873   cint 4404   class class class wbr 4577  dom cdm 5028  Oncon0 5626  cfv 5790  cen 7815  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-en 7819  df-card 8625
This theorem is referenced by:  cardid2  8639  oncardval  8641  cardidm  8645  cardne  8651  cardval  9224
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