Theorem rankid 4682

Description: Identity law for the rank function.

Hypothesis

Ref	Expression
rankid.1	⊢ A ∈ V

Assertion

Ref	Expression
rankid	⊢ A ∈ (R1 ‘suc (rank ‘A))

Proof of Theorem rankid

Step	Hyp	Ref	Expression
1		rankid.1	. . . 4 ⊢ A ∈ V
2		rankwflem 4675	. . . 4 ⊢ (A ∈ V → ∃x ∈ On A ∈ (R1 ‘suc x))
3	1, 2	ax-mp 7	. . 3 ⊢ ∃x ∈ On A ∈ (R1 ‘suc x)
4		ax-17 973	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
5		ax-17 973	. . . . . 6 ⊢ (y ∈ R1 → ∀x y ∈ R1)
6		hbrab1 1775	. . . . . . . 8 ⊢ (y ∈ {x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ {x ∈ On∣A ∈ (R1 ‘suc x)})
7	6	hbint 2547	. . . . . . 7 ⊢ (y ∈ ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
8	7	hbsuc 3046	. . . . . 6 ⊢ (y ∈ suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
9	5, 8	hbfv 3735	. . . . 5 ⊢ (y ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}) → ∀x y ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
10	4, 9	hbel 1569	. . . 4 ⊢ (A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}) → ∀x A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
11		suceq 3040	. . . . . 6 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → suc x = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
12	11	fveq2d 3734	. . . . 5 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → (R1 ‘suc x) = (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
13	12	eleq2d 1544	. . . 4 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → (A ∈ (R1 ‘suc x) ↔ A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})))
14	10, 13	onminsb 3015	. . 3 ⊢ (∃x ∈ On A ∈ (R1 ‘suc x) → A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
15	3, 14	ax-mp 7	. 2 ⊢ A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
16	1	rankval 4678	. . . 4 ⊢ (rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
17		suceq 3040	. . . 4 ⊢ ((rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → suc (rank ‘A) = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
18	16, 17	ax-mp 7	. . 3 ⊢ suc (rank ‘A) = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
19	18	fveq2i 3733	. 2 ⊢ (R1 ‘suc (rank ‘A)) = (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
20	15, 19	eleqtrr 1550	1 ⊢ A ∈ (R1 ‘suc (rank ‘A))

Syntax hints:   = wceq 958   ∈ wcel 960  ∃wrex 1649  {crab 1651  Vcvv 1814  ∩cint 2537  Oncon0 2954  suc csuc 2956   ‘cfv 3188  R₁cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankr1lem 4683  rankel 4690  rankval3 4691  bndrank 4692  rankpw 4694  rankval4 4712

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

Copyright terms: Public domain