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Theorem cardval3 9373
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 3511 . 2 (𝐴 ∈ dom card → 𝐴 ∈ V)
2 isnum2 9366 . . . 4 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
3 rabn0 4337 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 intex 5231 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
52, 3, 43bitr2i 301 . . 3 (𝐴 ∈ dom card ↔ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
65biimpi 218 . 2 (𝐴 ∈ dom card → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
7 breq2 5061 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rabbidv 3479 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
98inteqd 4872 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑥𝑦} = {𝑥 ∈ On ∣ 𝑥𝐴})
10 df-card 9360 . . 3 card = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑥𝑦})
119, 10fvmptg 6759 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V) → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
121, 6, 11syl2anc 586 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  wne 3014  wrex 3137  {crab 3140  Vcvv 3493  c0 4289   cint 4867   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cfv 6348  cen 8498  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-en 8502  df-card 9360
This theorem is referenced by:  cardid2  9374  oncardval  9376  cardidm  9380  cardne  9386  cardval  9960
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