Theorem tfr2b 7644

Description: Without assuming ax-rep 4902, 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 7134	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		eqid 2770	. . . . 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 7640	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
5	4	dmeqi 5463	. . . . 5 ⊢ dom 𝐹 = dom recs(𝐺)
6	5	eleq2i 2841	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺))
7	4	reseq1i 5530	. . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	eleq1i 2840	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
9	3, 6, 8	3bitr4g 303	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
10		onprc 7130	. . . . . 6 ⊢ ¬ On ∈ V
11		elex 3361	. . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V)
12	10, 11	mto 188	. . . . 5 ⊢ ¬ On ∈ dom 𝐹
13		eleq1 2837	. . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
14	12, 13	mtbiri 316	. . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
15	2	tfrlem13 7638	. . . . . 6 ⊢ ¬ recs(𝐺) ∈ V
16	4	eleq1i 2840	. . . . . 6 ⊢ (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
17	15, 16	mtbir 312	. . . . 5 ⊢ ¬ 𝐹 ∈ V
18		reseq2 5529	. . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On))
19	4	tfr1a 7642	. . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
20	19	simpli 470	. . . . . . . . 9 ⊢ Fun 𝐹
21		funrel 6048	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝐹
23	19	simpri 473	. . . . . . . . 9 ⊢ Lim dom 𝐹
24		limord 5927	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
25		ordsson 7135	. . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
26	23, 24, 25	mp2b 10	. . . . . . . 8 ⊢ dom 𝐹 ⊆ On
27		relssres 5578	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	22, 26, 27	mp2an 664	. . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹
29	18, 28	syl6eq 2820	. . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹)
30	29	eleq1d 2834	. . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V))
31	17, 30	mtbiri 316	. . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V)
32	14, 31	2falsed 365	. . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
33	9, 32	jaoi 837	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
34	1, 33	sylbi 207	1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 826   = wceq 1630   ∈ wcel 2144  {cab 2756  ∀wral 3060  ∃wrex 3061  Vcvv 3349   ⊆ wss 3721  dom cdm 5249   ↾ cres 5251  Rel wrel 5254  Ord word 5865  Oncon0 5866  Lim wlim 5867  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031  recscrecs 7619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7558  df-recs 7620

This theorem is referenced by:  ordtypelem3  8580  ordtypelem9  8586