Theorem iscard4 43774

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 5774	. . . . 5 ⊢ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
3		df-card 9851	. . . . . 6 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	releqi 5727	. . . . 5 ⊢ (Rel card ↔ Rel (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}))
5	2, 4	mpbir 231	. . . 4 ⊢ Rel card
6		relelrnb 5896	. . . 4 ⊢ (Rel card → (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran card ↔ ∃𝑥 𝑥card𝐴)
8	3	funmpt2 6531	. . . . . . 7 ⊢ Fun card
9		funbrfv 6882	. . . . . . 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 1836	. . . 4 ⊢ (∃𝑥 𝑥card𝐴 → ∃𝑥 𝐴 = (card‘𝑥))
13		cardidm 9871	. . . . . . 7 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
14		fveq2 6834	. . . . . . 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 1931	. . . . 5 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → (card‘𝐴) = 𝐴)
18	1	biimpi 216	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 = (card‘𝐴))
19		cardon 9856	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
20	18, 19	eqeltrdi 2844	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
21		onenon 9861	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
23		funfvbrb 6996	. . . . . . . . . 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 5124	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴card𝐴)
28		id 22	. . . . . . . . . 10 ⊢ (𝐴 = (card‘𝐴) → 𝐴 = (card‘𝐴))
29	28, 19	eqeltrdi 2844	. . . . . . . . 9 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ On)
30	29	eqcoms 2744	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
31		sbcbr1g 5155	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ([𝐴 / 𝑥]𝑥card𝐴 ↔ ⦋𝐴 / 𝑥⦌𝑥card𝐴))
32		csbvarg 4386	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
33	32	breq1d 5108	. . . . . . . . 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 3833	. . . . 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 9872	. . 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 1541  ∃wex 1780   ∈ wcel 2113  {crab 3399  Vcvv 3440  [wsbc 3740  ⦋csb 3849  ∩ cint 4902   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ran crn 5625  Rel wrel 5629  Oncon0 6317  Fun wfun 6486  ‘cfv 6492   ≈ cen 8880  cardccrd 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8635  df-en 8884  df-card 9851

This theorem is referenced by:  minregex  43775  minregex2  43776  elrncard  43778  alephiso2  43799