Theorem tz9.12 9743

Description: A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 9740 through tz9.12lem3 9742. (Contributed by NM, 22-Sep-2003.)

Hypothesis

Ref	Expression
tz9.12.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.12	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem tz9.12

Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tz9.12.1	. . . 4 ⊢ 𝐴 ∈ V
2		eqid 2729	. . . 4 ⊢ (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
3	1, 2	tz9.12lem2 9741	. . 3 ⊢ suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴) ∈ On
4	3	onsuci 7814	. 2 ⊢ suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴) ∈ On
5	1, 2	tz9.12lem3 9742	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴)))
6		fveq2 6858	. . . 4 ⊢ (𝑦 = suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴) → (𝑅1‘𝑦) = (𝑅1‘suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴)))
7	6	eleq2d 2814	. . 3 ⊢ (𝑦 = suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴) → (𝐴 ∈ (𝑅1‘𝑦) ↔ 𝐴 ∈ (𝑅1‘suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴))))
8	7	rspcev 3588	. 2 ⊢ ((suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴) ∈ On ∧ 𝐴 ∈ (𝑅1‘suc suc ∪ ((𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) “ 𝐴))) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))
9	4, 5, 8	sylancr 587	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447  ∪ cuni 4871  ∩ cint 4910   ↦ cmpt 5188   “ cima 5641  Oncon0 6332  suc csuc 6334  ‘cfv 6511  𝑅₁cr1 9715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717

This theorem is referenced by:  tz9.13  9744  r1elss  9759