Theorem iscard4 44795

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 2770	. 2 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 = (card‘𝐴))
2		mptrel 5813	. . . . 5 ⊢ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
3		df-card 10057	. . . . . 6 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	releqi 5762	. . . . 5 ⊢ (Rel card ↔ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}))
5	2, 4	mpbir 234	. . . 4 ⊢ Rel card
6		relelrnb 5941	. . . 4 ⊢ (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
8	3	funmpt2 6607	. . . . . . 7 ⊢ Fun card
9		funbrfv 6964	. . . . . . 7 ⊢ (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (𝑥card𝐴 → (card‘𝑥) = 𝐴)
11	10	eqcomd 2769	. . . . 5 ⊢ (𝑥card𝐴 → 𝐴 = (card‘𝑥))
12	11	eximi 1868	. . . 4 ⊢ (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13		cardidm 10077	. . . . . . 7 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
14		fveq2 6916	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15		id 23	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
16	13, 14, 15	3eqtr4a 2824	. . . . . 6 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
17	16	exlimiv 1965	. . . . 5 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
18	1	biimpi 219	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 = (card‘𝐴))
19		cardon 10062	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
20	18, 19	eqeltrdi 2872	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
21		onenon 10067	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
22	20, 21	syl 18	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
23		funfvbrb 7081	. . . . . . . . . 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 5137	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card𝐴)
28		id 23	. . . . . . . . . 10 ⊢ (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
29	28, 19	eqeltrdi 2872	. . . . . . . . 9 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
30	29	eqcoms 2771	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
31		sbcbr1g 5169	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ ⦋𝐴 / 𝑥⦌𝑥card𝐴))
32		csbvarg 4395	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
33	32	breq1d 5118	. . . . . . . . 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 3834	. . . . 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 10078	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416  Vcvv 3454  [wsbc 3742  ⦋csb 3851  ∩ cint 4912   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659  Rel wrel 5663  Oncon0 6388  Fun wfun 6560  ‘cfv 6566   ≈ cen 9008  cardccrd 10053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-er 8755  df-en 9012  df-card 10057

This theorem is used by:  minregex  44796  minregex2  44797  elrncard  44799  alephiso2  44820