Theorem tfrlem14 8380

Description: Lemma for transfinite recursion. Assuming ax-rep 5241, dom recs ∈ V ↔ recs ∈ V, so since dom recs is an ordinal, it must be equal to On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem14	⊢ dom recs(𝐹) = On

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)

Proof of Theorem tfrlem14

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem13 8379	. . 3 ⊢ ¬ recs(𝐹) ∈ V
3	1	tfrlem7 8372	. . . 4 ⊢ Fun recs(𝐹)
4		funex 7221	. . . 4 ⊢ ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V)
5	3, 4	mpan 703	. . 3 ⊢ (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V)
6	2, 5	mto 200	. 2 ⊢ ¬ dom recs(𝐹) ∈ On
7	1	tfrlem8 8373	. . 3 ⊢ Ord dom recs(𝐹)
8		ordeleqon 7783	. . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On))
9	7, 8	mpbi 233	. 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)
10	6, 9	mtpor 1803	1 ⊢ dom recs(𝐹) = On

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092  Vcvv 3458  dom cdm 5664   ↾ cres 5666  Ord word 6363  Oncon0 6364  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540  recscrecs 8359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-tr 5222  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360

This theorem is used by:  tfr1  8386