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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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