Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscard4 Structured version   Visualization version   GIF version

Theorem iscard4 42284
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 2740 . 2 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 = (cardβ€˜π΄))
2 mptrel 5826 . . . . 5 Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
3 df-card 9934 . . . . . 6 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
43releqi 5778 . . . . 5 (Rel card ↔ Rel (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯}))
52, 4mpbir 230 . . . 4 Rel card
6 relelrnb 5947 . . . 4 (Rel card β†’ (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran card ↔ βˆƒπ‘₯ π‘₯card𝐴)
83funmpt2 6588 . . . . . . 7 Fun card
9 funbrfv 6943 . . . . . . 7 (Fun card β†’ (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴))
108, 9ax-mp 5 . . . . . 6 (π‘₯card𝐴 β†’ (cardβ€˜π‘₯) = 𝐴)
1110eqcomd 2739 . . . . 5 (π‘₯card𝐴 β†’ 𝐴 = (cardβ€˜π‘₯))
1211eximi 1838 . . . 4 (βˆƒπ‘₯ π‘₯card𝐴 β†’ βˆƒπ‘₯ 𝐴 = (cardβ€˜π‘₯))
13 cardidm 9954 . . . . . . 7 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
14 fveq2 6892 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π‘₯)))
15 id 22 . . . . . . 7 (𝐴 = (cardβ€˜π‘₯) β†’ 𝐴 = (cardβ€˜π‘₯))
1613, 14, 153eqtr4a 2799 . . . . . 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 9939 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
2018, 19eqeltrdi 2842 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
21 onenon 9944 . . . . . . . . . 10 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
2220, 21syl 17 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
23 funfvbrb 7053 . . . . . . . . . 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 5175 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴card𝐴)
28 id 22 . . . . . . . . . 10 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 = (cardβ€˜π΄))
2928, 19eqeltrdi 2842 . . . . . . . . 9 (𝐴 = (cardβ€˜π΄) β†’ 𝐴 ∈ On)
3029eqcoms 2741 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
31 sbcbr1g 5206 . . . . . . . . 9 (𝐴 ∈ On β†’ ([𝐴 / π‘₯]π‘₯card𝐴 ↔ ⦋𝐴 / π‘₯⦌π‘₯card𝐴))
32 csbvarg 4432 . . . . . . . . . 10 (𝐴 ∈ On β†’ ⦋𝐴 / π‘₯⦌π‘₯ = 𝐴)
3332breq1d 5159 . . . . . . . . 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 3878 . . . . 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 9955 . . 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 3433  Vcvv 3475  [wsbc 3778  β¦‹csb 3894  βˆ© cint 4951   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  Rel wrel 5682  Oncon0 6365  Fun wfun 6538  β€˜cfv 6544   β‰ˆ cen 8936  cardccrd 9930
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-card 9934
This theorem is referenced by:  minregex  42285  minregex2  42286  elrncard  42288  alephiso2  42309
  Copyright terms: Public domain W3C validator