Theorem rankvalb 9814

Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9833 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅₁. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Assertion

Ref	Expression
rankvalb	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankvalb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rank 9782	. 2 ⊢ rank = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)})
2		eleq1 2817	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝑥)))
3	2	rabbidv 3436	. . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
4	3	inteqd 4949	. 2 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
5		elex 3489	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V)
6		rankwflemb 9810	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
7		intexrab 5336	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
8	6, 7	sylbb 218	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
9	1, 4, 5, 8	fvmptd3 7022	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∃wrex 3066  {crab 3428  Vcvv 3470  ∪ cuni 4903  ∩ cint 4944   “ cima 5675  Oncon0 6363  suc csuc 6365  ‘cfv 6542  𝑅₁cr1 9779  rankcrnk 9780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9781  df-rank 9782

This theorem is referenced by:  rankr1ai  9815  rankidb  9817  rankval  9833