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Theorem iscard4 40228
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 2808 . 2 ((card‘𝐴) = 𝐴𝐴 = (card‘𝐴))
2 mptrel 5665 . . . . 5 Rel (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
3 df-card 9356 . . . . . 6 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
43releqi 5620 . . . . 5 (Rel card ↔ Rel (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
52, 4mpbir 234 . . . 4 Rel card
6 relelrnb 5785 . . . 4 (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
83funmpt2 6367 . . . . . . 7 Fun card
9 funbrfv 6695 . . . . . . 7 (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
108, 9ax-mp 5 . . . . . 6 (𝑥card𝐴 → (card‘𝑥) = 𝐴)
1110eqcomd 2807 . . . . 5 (𝑥card𝐴𝐴 = (card‘𝑥))
1211eximi 1836 . . . 4 (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13 cardidm 9376 . . . . . . 7 (card‘(card‘𝑥)) = (card‘𝑥)
14 fveq2 6649 . . . . . . 7 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15 id 22 . . . . . . 7 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
1613, 14, 153eqtr4a 2862 . . . . . 6 (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
1716exlimiv 1931 . . . . 5 (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
181biimpi 219 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 = (card‘𝐴))
19 cardon 9361 . . . . . . . . . . 11 (card‘𝐴) ∈ On
2018, 19eqeltrdi 2901 . . . . . . . . . 10 ((card‘𝐴) = 𝐴𝐴 ∈ On)
21 onenon 9366 . . . . . . . . . 10 (𝐴 ∈ On → 𝐴 ∈ dom card)
2220, 21syl 17 . . . . . . . . 9 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
23 funfvbrb 6802 . . . . . . . . . 10 (Fun card → (𝐴 ∈ dom card ↔ 𝐴card(card‘𝐴)))
2423biimpd 232 . . . . . . . . 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 5059 . . . . . . 7 ((card‘𝐴) = 𝐴𝐴card𝐴)
28 id 22 . . . . . . . . . 10 (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
2928, 19eqeltrdi 2901 . . . . . . . . 9 (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
3029eqcoms 2809 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
31 sbcbr1g 5090 . . . . . . . . 9 (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴𝐴 / 𝑥𝑥card𝐴))
32 csbvarg 4342 . . . . . . . . . 10 (𝐴 ∈ On → 𝐴 / 𝑥𝑥 = 𝐴)
3332breq1d 5043 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 / 𝑥𝑥card𝐴𝐴card𝐴))
3431, 33bitrd 282 . . . . . . . 8 (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴𝐴card𝐴))
3530, 34syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → ([𝐴 / 𝑥]𝑥card𝐴𝐴card𝐴))
3627, 35mpbird 260 . . . . . 6 ((card‘𝐴) = 𝐴[𝐴 / 𝑥]𝑥card𝐴)
3736spesbcd 3815 . . . . 5 ((card‘𝐴) = 𝐴 → ∃𝑥 𝑥card𝐴)
3817, 37syl 17 . . . 4 (∃𝑥 𝐴 = (card‘𝑥) → ∃𝑥 𝑥card𝐴)
3912, 38impbii 212 . . 3 (∃𝑥 𝑥card𝐴 ↔ ∃𝑥 𝐴 = (card‘𝑥))
40 oncard 9377 . . 3 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
417, 39, 403bitrri 301 . 2 (𝐴 = (card‘𝐴) ↔ 𝐴 ∈ ran card)
421, 41bitri 278 1 ((card‘𝐴) = 𝐴𝐴 ∈ ran card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wex 1781  wcel 2112  {crab 3113  Vcvv 3444  [wsbc 3723  csb 3831   cint 4841   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524  Rel wrel 5528  Oncon0 6163  Fun wfun 6322  cfv 6328  cen 8493  cardccrd 9352
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-er 8276  df-en 8497  df-card 9356
This theorem is referenced by:  elrncard  40230  alephiso2  40244
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