Theorem iscard4 43524

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 2743	. 2 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 = (card‘𝐴))
2		mptrel 5809	. . . . 5 ⊢ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
3		df-card 9958	. . . . . 6 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	releqi 5761	. . . . 5 ⊢ (Rel card ↔ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}))
5	2, 4	mpbir 231	. . . 4 ⊢ Rel card
6		relelrnb 5932	. . . 4 ⊢ (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
8	3	funmpt2 6580	. . . . . . 7 ⊢ Fun card
9		funbrfv 6932	. . . . . . 7 ⊢ (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (𝑥card𝐴 → (card‘𝑥) = 𝐴)
11	10	eqcomd 2742	. . . . 5 ⊢ (𝑥card𝐴 → 𝐴 = (card‘𝑥))
12	11	eximi 1835	. . . 4 ⊢ (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13		cardidm 9978	. . . . . . 7 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
14		fveq2 6881	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15		id 22	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
16	13, 14, 15	3eqtr4a 2797	. . . . . 6 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
17	16	exlimiv 1930	. . . . 5 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
18	1	biimpi 216	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 = (card‘𝐴))
19		cardon 9963	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
20	18, 19	eqeltrdi 2843	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
21		onenon 9968	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
23		funfvbrb 7046	. . . . . . . . . 10 ⊢ (Fun card → (𝐴 ∈ dom card ↔ 𝐴card(card‘𝐴)))
24	23	biimpd 229	. . . . . . . . 9 ⊢ (Fun card → (𝐴 ∈ dom card → 𝐴card(card‘𝐴)))
25	8, 22, 24	mpsyl 68	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card(card‘𝐴))
26		id 22	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
27	25, 26	breqtrd 5150	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card𝐴)
28		id 22	. . . . . . . . . 10 ⊢ (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
29	28, 19	eqeltrdi 2843	. . . . . . . . 9 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
30	29	eqcoms 2744	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
31		sbcbr1g 5181	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ ⦋𝐴 / 𝑥⦌𝑥card𝐴))
32		csbvarg 4414	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
33	32	breq1d 5134	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (⦋𝐴 / 𝑥⦌𝑥card𝐴 ↔ 𝐴card𝐴))
34	31, 33	bitrd 279	. . . . . . . 8 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ 𝐴card𝐴))
35	30, 34	syl 17	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → ([𝐴 / 𝑥]𝑥card𝐴 ↔ 𝐴card𝐴))
36	27, 35	mpbird 257	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → [𝐴 / 𝑥]𝑥card𝐴)
37	36	spesbcd 3863	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → ∃𝑥 𝑥card𝐴)
38	17, 37	syl 17	. . . 4 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → ∃𝑥 𝑥card𝐴)
39	12, 38	impbii 209	. . 3 ⊢ (∃𝑥 𝑥card𝐴 ↔ ∃𝑥 𝐴 = (card‘𝑥))
40		oncard 9979	. . 3 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
41	7, 39, 40	3bitrri 298	. 2 ⊢ (𝐴 = (card‘𝐴) ↔ 𝐴 ∈ ran card)
42	1, 41	bitri 275	1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {crab 3420  Vcvv 3464  [wsbc 3770  ⦋csb 3879  ∩ cint 4927   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659  ran crn 5660  Rel wrel 5664  Oncon0 6357  Fun wfun 6530  ‘cfv 6536   ≈ cen 8961  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8724  df-en 8965  df-card 9958

This theorem is referenced by:  minregex  43525  minregex2  43526  elrncard  43528  alephiso2  43549