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Theorem iscard4 41812
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 2744 . 2 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 = (cardβ€˜π΄))
2 mptrel 5782 . . . . 5 Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
3 df-card 9876 . . . . . 6 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
43releqi 5734 . . . . 5 (Rel card ↔ Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯}))
52, 4mpbir 230 . . . 4 Rel card
6 relelrnb 5903 . . . 4 (Rel card β†’ (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴)
83funmpt2 6541 . . . . . . 7 Fun card
9 funbrfv 6894 . . . . . . 7 (Fun card β†’ (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴))
108, 9ax-mp 5 . . . . . 6 (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴)
1110eqcomd 2743 . . . . 5 (π‘₯card𝐴 β†’ 𝐴 = (cardβ€˜π‘₯))
1211eximi 1838 . . . 4 (βˆƒπ‘₯ π‘₯card𝐴 β†’ βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯))
13 cardidm 9896 . . . . . . 7 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
14 fveq2 6843 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π‘₯)))
15 id 22 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ 𝐴 = (cardβ€˜π‘₯))
1613, 14, 153eqtr4a 2803 . . . . . 6 (𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
1716exlimiv 1934 . . . . 5 (βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
181biimpi 215 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 = (cardβ€˜π΄))
19 cardon 9881 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
2018, 19eqeltrdi 2846 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
21 onenon 9886 . . . . . . . . . 10 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
2220, 21syl 17 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
23 funfvbrb 7002 . . . . . . . . . 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 5132 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴card𝐴)
28 id 22 . . . . . . . . . 10 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 = (cardβ€˜π΄))
2928, 19eqeltrdi 2846 . . . . . . . . 9 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 ∈ On)
3029eqcoms 2745 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
31 sbcbr1g 5163 . . . . . . . . 9 (𝐴 ∈ On β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ ⦋𝐴 / π‘₯⦌π‘₯card𝐴))
32 csbvarg 4392 . . . . . . . . . 10 (𝐴 ∈ On β†’ ⦋𝐴 / π‘₯⦌π‘₯ = 𝐴)
3332breq1d 5116 . . . . . . . . 9 (𝐴 ∈ On β†’ (⦋𝐴 / π‘₯⦌π‘₯card𝐴 ↔ 𝐴card𝐴))
3431, 33bitrd 279 . . . . . . . 8 (𝐴 ∈ On β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ 𝐴card𝐴))
3530, 34syl 17 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ 𝐴card𝐴))
3627, 35mpbird 257 . . . . . 6 ((cardβ€˜π΄) = 𝐴 β†’ [𝐴 / π‘₯]π‘₯card𝐴)
3736spesbcd 3840 . . . . 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 9897 . . 3 (βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯) ↔ 𝐴 = (cardβ€˜π΄))
417, 39, 403bitrri 298 . 2 (𝐴 = (cardβ€˜π΄) ↔ 𝐴 ∈ ran card)
421, 41bitri 275 1 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {crab 3408  Vcvv 3446  [wsbc 3740  β¦‹csb 3856  βˆ© cint 4908   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634  ran crn 5635  Rel wrel 5639  Oncon0 6318  Fun wfun 6491  β€˜cfv 6497   β‰ˆ cen 8881  cardccrd 9872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8649  df-en 8885  df-card 9876
This theorem is referenced by:  minregex  41813  minregex2  41814  elrncard  41816  alephiso2  41837
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