Theorem iscard4 44151

Description: Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)

Assertion

Ref	Expression
iscard4	⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)

Proof of Theorem iscard4

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2776	. 2 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 = (card‘𝐴))
2		mptrel 5813	. . . . 5 ⊢ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
3		df-card 9925	. . . . . 6 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	releqi 5765	. . . . 5 ⊢ (Rel card ↔ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}))
5	2, 4	mpbir 234	. . . 4 ⊢ Rel card
6		relelrnb 5938	. . . 4 ⊢ (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
8	3	funmpt2 6576	. . . . . . 7 ⊢ Fun card
9		funbrfv 6930	. . . . . . 7 ⊢ (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (𝑥card𝐴 → (card‘𝑥) = 𝐴)
11	10	eqcomd 2775	. . . . 5 ⊢ (𝑥card𝐴 → 𝐴 = (card‘𝑥))
12	11	eximi 1862	. . . 4 ⊢ (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13		cardidm 9945	. . . . . . 7 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
14		fveq2 6882	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15		id 23	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
16	13, 14, 15	3eqtr4a 2830	. . . . . 6 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
17	16	exlimiv 1957	. . . . 5 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
18	1	biimpi 219	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 = (card‘𝐴))
19		cardon 9930	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
20	18, 19	eqeltrdi 2877	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
21		onenon 9935	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
22	20, 21	syl 18	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
23		funfvbrb 7047	. . . . . . . . . 10 ⊢ (Fun card → (𝐴 ∈ dom card ↔ 𝐴card(card‘𝐴)))
24	23	biimpd 232	. . . . . . . . 9 ⊢ (Fun card → (𝐴 ∈ dom card → 𝐴card(card‘𝐴)))
25	8, 22, 24	mpsyl 69	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card(card‘𝐴))
26		id 23	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
27	25, 26	breqtrd 5141	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card𝐴)
28		id 23	. . . . . . . . . 10 ⊢ (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
29	28, 19	eqeltrdi 2877	. . . . . . . . 9 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
30	29	eqcoms 2777	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
31		sbcbr1g 5172	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ ⦋𝐴 / 𝑥⦌𝑥card𝐴))
32		csbvarg 4405	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
33	32	breq1d 5123	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (⦋𝐴 / 𝑥⦌𝑥card𝐴 ↔ 𝐴card𝐴))
34	31, 33	bitrd 282	. . . . . . . 8 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ 𝐴card𝐴))
35	30, 34	syl 18	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → ([𝐴 / 𝑥]𝑥card𝐴 ↔ 𝐴card𝐴))
36	27, 35	mpbird 260	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → [𝐴 / 𝑥]𝑥card𝐴)
37	36	spesbcd 3845	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → ∃𝑥 𝑥card𝐴)
38	17, 37	syl 18	. . . 4 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → ∃𝑥 𝑥card𝐴)
39	12, 38	impbii 212	. . 3 ⊢ (∃𝑥 𝑥card𝐴 ↔ ∃𝑥 𝐴 = (card‘𝑥))
40		oncard 9946	. . 3 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
41	7, 39, 40	3bitrri 301	. 2 ⊢ (𝐴 = (card‘𝐴) ↔ 𝐴 ∈ ran card)
42	1, 41	bitri 278	1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {crab 3423  Vcvv 3463  [wsbc 3753  ⦋csb 3861  ∩ cint 4916   class class class wbr 5113   ↦ cmpt 5196  dom cdm 5662  ran crn 5663  Rel wrel 5667  Oncon0 6361  Fun wfun 6531  ‘cfv 6537   ≈ cen 8940  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8694  df-en 8944  df-card 9925

This theorem is referenced by:  minregex  44152  minregex2  44153  elrncard  44155  alephiso2  44176