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Theorem cardval3 9455
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 3416 . 2 (𝐴 ∈ dom card → 𝐴 ∈ V)
2 isnum2 9448 . . . 4 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
3 rabn0 4275 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 intex 5206 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
52, 3, 43bitr2i 302 . . 3 (𝐴 ∈ dom card ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
65biimpi 219 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
7 breq2 5035 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rabbidv 3381 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
98inteqd 4842 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
10 df-card 9442 . . 3 card = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑥𝑦})
119, 10fvmptg 6774 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V) → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
121, 6, 11syl2anc 587 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  wne 2934  wrex 3054  {crab 3057  Vcvv 3398  c0 4212   cint 4837   class class class wbr 5031  dom cdm 5526  Oncon0 6173  cfv 6340  cen 8553  cardccrd 9438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-en 8557  df-card 9442
This theorem is referenced by:  cardid2  9456  oncardval  9458  cardidm  9462  cardne  9468  cardval  10047
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