Theorem rankidb 8546

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Assertion

Ref	Expression
rankidb	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Proof of Theorem rankidb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankwflemb 8539	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝑅1
3		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
4	3	nfint 4421	. . . . . . 7 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
5	4	nfsuc 5713	. . . . . 6 ⊢ Ⅎ𝑥 suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
6	2, 5	nffv 6110	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
7	6	nfel2 2767	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8		suceq 5707	. . . . . 6 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
9	8	fveq2d 6107	. . . . 5 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
10	9	eleq2d 2673	. . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
11	7, 10	onminsb 6891	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
12	1, 11	sylbi 206	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13		rankvalb 8543	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14		suceq 5707	. . . 4 ⊢ ((rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
15	13, 14	syl 17	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
16	15	fveq2d 6107	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
17	12, 16	eleqtrrd 2691	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  ∪ cuni 4372  ∩ cint 4410   “ cima 5041  Oncon0 5640  suc csuc 5642  ‘cfv 5804  𝑅₁cr1 8508  rankcrnk 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511

This theorem is referenced by:  rankdmr1  8547  rankr1ag  8548  sswf  8554  uniwf  8565  rankonidlem  8574  rankid  8579  dfac12lem2  8849  aomclem4  36645