Theorem tfr2b 8385

Description: Without assuming ax-rep 5240, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypothesis

Ref	Expression
tfr.1	⊢ 𝐹 = recs(𝐺)

Assertion

Ref	Expression
tfr2b	⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

Proof of Theorem tfr2b

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordeleqon 7783	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		eqid 2765	. . . . 5 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
3	2	tfrlem15 8381	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
5	4	dmeqi 5896	. . . . 5 ⊢ dom 𝐹 = dom recs(𝐺)
6	5	eleq2i 2857	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺))
7	4	reseq1i 5976	. . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	eleq1i 2856	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
9	3, 6, 8	3bitr4g 317	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
10		onprc 7779	. . . . . 6 ⊢ ¬ On ∈ V
11		elex 3478	. . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V)
12	10, 11	mto 200	. . . . 5 ⊢ ¬ On ∈ dom 𝐹
13		eleq1 2853	. . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
14	12, 13	mtbiri 330	. . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
15	2	tfrlem13 8379	. . . . . 6 ⊢ ¬ recs(𝐺) ∈ V
16	4, 15	eqneltri 2884	. . . . 5 ⊢ ¬ 𝐹 ∈ V
17		reseq2 5975	. . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On))
18	4	tfr1a 8383	. . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
19	18	simpli 489	. . . . . . . . 9 ⊢ Fun 𝐹
20		funrel 6557	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝐹
22	18	simpri 491	. . . . . . . . 9 ⊢ Lim dom 𝐹
23		limord 6426	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
24		ordsson 7784	. . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
25	22, 23, 24	mp2b 10	. . . . . . . 8 ⊢ dom 𝐹 ⊆ On
26		relssres 6023	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
27	21, 25, 26	mp2an 705	. . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹
28	17, 27	eqtrdi 2816	. . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹)
29	28	eleq1d 2850	. . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V))
30	16, 29	mtbiri 330	. . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V)
31	14, 30	2falsed 379	. . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
32	9, 31	jaoi 871	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
33	1, 32	sylbi 220	1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  dom cdm 5663   ↾ cres 5665  Rel wrel 5668  Ord word 6363  Oncon0 6364  Lim wlim 6365  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 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  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:  ordtypelem3  9485  ordtypelem9  9491