Theorem iscard4 43981

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 2744	. 2 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 = (card‘𝐴))
2		mptrel 5775	. . . . 5 ⊢ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
3		df-card 9857	. . . . . 6 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	releqi 5728	. . . . 5 ⊢ (Rel card ↔ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}))
5	2, 4	mpbir 231	. . . 4 ⊢ Rel card
6		relelrnb 5897	. . . 4 ⊢ (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
8	3	funmpt2 6532	. . . . . . 7 ⊢ Fun card
9		funbrfv 6883	. . . . . . 7 ⊢ (Fun card → (𝑥card𝐴 → (card‘𝑥) = 𝐴))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (𝑥card𝐴 → (card‘𝑥) = 𝐴)
11	10	eqcomd 2743	. . . . 5 ⊢ (𝑥card𝐴 → 𝐴 = (card‘𝑥))
12	11	eximi 1837	. . . 4 ⊢ (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13		cardidm 9877	. . . . . . 7 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
14		fveq2 6835	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
15		id 22	. . . . . . 7 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
16	13, 14, 15	3eqtr4a 2798	. . . . . 6 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
17	16	exlimiv 1932	. . . . 5 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
18	1	biimpi 216	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 = (card‘𝐴))
19		cardon 9862	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
20	18, 19	eqeltrdi 2845	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
21		onenon 9867	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
23		funfvbrb 6998	. . . . . . . . . 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 5112	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card𝐴)
28		id 22	. . . . . . . . . 10 ⊢ (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
29	28, 19	eqeltrdi 2845	. . . . . . . . 9 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
30	29	eqcoms 2745	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
31		sbcbr1g 5143	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ ⦋𝐴 / 𝑥⦌𝑥card𝐴))
32		csbvarg 4375	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
33	32	breq1d 5096	. . . . . . . . 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 3822	. . . . 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 9878	. . 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 1542  ∃wex 1781   ∈ wcel 2114  {crab 3390  Vcvv 3430  [wsbc 3729  ⦋csb 3838  ∩ cint 4890   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5625  ran crn 5626  Rel wrel 5630  Oncon0 6318  Fun wfun 6487  ‘cfv 6493   ≈ cen 8884  cardccrd 9853

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-er 8637  df-en 8888  df-card 9857

This theorem is referenced by:  minregex  43982  minregex2  43983  elrncard  43985  alephiso2  44006