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Theorem iscard4 41710
Description: Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
Assertion
Ref Expression
iscard4 ((card‘𝐴) = 𝐴𝐴 ∈ ran card)

Proof of Theorem iscard4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2743 . 2 ((card‘𝐴) = 𝐴𝐴 = (card‘𝐴))
2 mptrel 5779 . . . . 5 Rel (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
3 df-card 9833 . . . . . 6 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
43releqi 5731 . . . . 5 (Rel card ↔ Rel (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
52, 4mpbir 230 . . . 4 Rel card
6 relelrnb 5900 . . . 4 (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
83funmpt2 6537 . . . . . . 7 Fun card
9 funbrfv 6890 . . . . . . 7 (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
108, 9ax-mp 5 . . . . . 6 (𝑥card𝐴 → (card‘𝑥) = 𝐴)
1110eqcomd 2742 . . . . 5 (𝑥card𝐴𝐴 = (card‘𝑥))
1211eximi 1837 . . . 4 (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13 cardidm 9853 . . . . . . 7 (card‘(card‘𝑥)) = (card‘𝑥)
14 fveq2 6839 . . . . . . 7 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15 id 22 . . . . . . 7 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
1613, 14, 153eqtr4a 2802 . . . . . 6 (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
1716exlimiv 1933 . . . . 5 (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
181biimpi 215 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 = (card‘𝐴))
19 cardon 9838 . . . . . . . . . . 11 (card‘𝐴) ∈ On
2018, 19eqeltrdi 2846 . . . . . . . . . 10 ((card‘𝐴) = 𝐴𝐴 ∈ On)
21 onenon 9843 . . . . . . . . . 10 (𝐴 ∈ On → 𝐴 ∈ dom card)
2220, 21syl 17 . . . . . . . . 9 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
23 funfvbrb 6998 . . . . . . . . . 10 (Fun card → (𝐴 ∈ dom card ↔ 𝐴card(card‘𝐴)))
2423biimpd 228 . . . . . . . . 9 (Fun card → (𝐴 ∈ dom card → 𝐴card(card‘𝐴)))
258, 22, 24mpsyl 68 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴card(card‘𝐴))
26 id 22 . . . . . . . 8 ((card‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
2725, 26breqtrd 5129 . . . . . . 7 ((card‘𝐴) = 𝐴𝐴card𝐴)
28 id 22 . . . . . . . . . 10 (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
2928, 19eqeltrdi 2846 . . . . . . . . 9 (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
3029eqcoms 2744 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
31 sbcbr1g 5160 . . . . . . . . 9 (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴𝐴 / 𝑥𝑥card𝐴))
32 csbvarg 4389 . . . . . . . . . 10 (𝐴 ∈ On → 𝐴 / 𝑥𝑥 = 𝐴)
3332breq1d 5113 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 / 𝑥𝑥card𝐴𝐴card𝐴))
3431, 33bitrd 278 . . . . . . . 8 (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴𝐴card𝐴))
3530, 34syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → ([𝐴 / 𝑥]𝑥card𝐴𝐴card𝐴))
3627, 35mpbird 256 . . . . . 6 ((card‘𝐴) = 𝐴[𝐴 / 𝑥]𝑥card𝐴)
3736spesbcd 3837 . . . . 5 ((card‘𝐴) = 𝐴 → ∃𝑥 𝑥card𝐴)
3817, 37syl 17 . . . 4 (∃𝑥 𝐴 = (card‘𝑥) → ∃𝑥 𝑥card𝐴)
3912, 38impbii 208 . . 3 (∃𝑥 𝑥card𝐴 ↔ ∃𝑥 𝐴 = (card‘𝑥))
40 oncard 9854 . . 3 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
417, 39, 403bitrri 297 . 2 (𝐴 = (card‘𝐴) ↔ 𝐴 ∈ ran card)
421, 41bitri 274 1 ((card‘𝐴) = 𝐴𝐴 ∈ ran card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wex 1781  wcel 2106  {crab 3405  Vcvv 3443  [wsbc 3737  csb 3853   cint 4905   class class class wbr 5103  cmpt 5186  dom cdm 5631  ran crn 5632  Rel wrel 5636  Oncon0 6315  Fun wfun 6487  cfv 6493  cen 8838  cardccrd 9829
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 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6318  df-on 6319  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-er 8606  df-en 8842  df-card 9833
This theorem is referenced by:  minregex  41711  minregex2  41712  elrncard  41714  alephiso2  41735
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