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Theorem iscard4 42269
Description: Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
Assertion
Ref Expression
iscard4 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)

Proof of Theorem iscard4
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2739 . 2 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 = (cardβ€˜π΄))
2 mptrel 5823 . . . . 5 Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
3 df-card 9930 . . . . . 6 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
43releqi 5775 . . . . 5 (Rel card ↔ Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯}))
52, 4mpbir 230 . . . 4 Rel card
6 relelrnb 5944 . . . 4 (Rel card β†’ (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴)
83funmpt2 6584 . . . . . . 7 Fun card
9 funbrfv 6939 . . . . . . 7 (Fun card β†’ (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴))
108, 9ax-mp 5 . . . . . 6 (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴)
1110eqcomd 2738 . . . . 5 (π‘₯card𝐴 β†’ 𝐴 = (cardβ€˜π‘₯))
1211eximi 1837 . . . 4 (βˆƒπ‘₯ π‘₯card𝐴 β†’ βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯))
13 cardidm 9950 . . . . . . 7 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
14 fveq2 6888 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π‘₯)))
15 id 22 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ 𝐴 = (cardβ€˜π‘₯))
1613, 14, 153eqtr4a 2798 . . . . . 6 (𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
1716exlimiv 1933 . . . . 5 (βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
181biimpi 215 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 = (cardβ€˜π΄))
19 cardon 9935 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
2018, 19eqeltrdi 2841 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
21 onenon 9940 . . . . . . . . . 10 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
2220, 21syl 17 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
23 funfvbrb 7049 . . . . . . . . . 10 (Fun card β†’ (𝐴 ∈ dom card ↔ 𝐴card(cardβ€˜π΄)))
2423biimpd 228 . . . . . . . . 9 (Fun card β†’ (𝐴 ∈ dom card β†’ 𝐴card(cardβ€˜π΄)))
258, 22, 24mpsyl 68 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴card(cardβ€˜π΄))
26 id 22 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ (cardβ€˜π΄) = 𝐴)
2725, 26breqtrd 5173 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴card𝐴)
28 id 22 . . . . . . . . . 10 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 = (cardβ€˜π΄))
2928, 19eqeltrdi 2841 . . . . . . . . 9 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 ∈ On)
3029eqcoms 2740 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
31 sbcbr1g 5204 . . . . . . . . 9 (𝐴 ∈ On β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ ⦋𝐴 / π‘₯⦌π‘₯card𝐴))
32 csbvarg 4430 . . . . . . . . . 10 (𝐴 ∈ On β†’ ⦋𝐴 / π‘₯⦌π‘₯ = 𝐴)
3332breq1d 5157 . . . . . . . . 9 (𝐴 ∈ On β†’ (⦋𝐴 / π‘₯⦌π‘₯card𝐴 ↔ 𝐴card𝐴))
3431, 33bitrd 278 . . . . . . . 8 (𝐴 ∈ On β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ 𝐴card𝐴))
3530, 34syl 17 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ 𝐴card𝐴))
3627, 35mpbird 256 . . . . . 6 ((cardβ€˜π΄) = 𝐴 β†’ [𝐴 / π‘₯]π‘₯card𝐴)
3736spesbcd 3876 . . . . 5 ((cardβ€˜π΄) = 𝐴 β†’ βˆƒπ‘₯ π‘₯card𝐴)
3817, 37syl 17 . . . 4 (βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯) β†’ βˆƒπ‘₯ π‘₯card𝐴)
3912, 38impbii 208 . . 3 (βˆƒπ‘₯ π‘₯card𝐴 ↔ βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯))
40 oncard 9951 . . 3 (βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯) ↔ 𝐴 = (cardβ€˜π΄))
417, 39, 403bitrri 297 . 2 (𝐴 = (cardβ€˜π΄) ↔ 𝐴 ∈ ran card)
421, 41bitri 274 1 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {crab 3432  Vcvv 3474  [wsbc 3776  β¦‹csb 3892  βˆ© cint 4949   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  Rel wrel 5680  Oncon0 6361  Fun wfun 6534  β€˜cfv 6540   β‰ˆ cen 8932  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-card 9930
This theorem is referenced by:  minregex  42270  minregex2  42271  elrncard  42273  alephiso2  42294
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