Theorem tfr2b 8325

Description: Without assuming ax-rep 5221, 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 7722	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		eqid 2729	. . . . 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 8321	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
5	4	dmeqi 5851	. . . . 5 ⊢ dom 𝐹 = dom recs(𝐺)
6	5	eleq2i 2820	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺))
7	4	reseq1i 5930	. . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	eleq1i 2819	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
9	3, 6, 8	3bitr4g 314	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
10		onprc 7718	. . . . . 6 ⊢ ¬ On ∈ V
11		elex 3459	. . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V)
12	10, 11	mto 197	. . . . 5 ⊢ ¬ On ∈ dom 𝐹
13		eleq1 2816	. . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
14	12, 13	mtbiri 327	. . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
15	2	tfrlem13 8319	. . . . . 6 ⊢ ¬ recs(𝐺) ∈ V
16	4, 15	eqneltri 2847	. . . . 5 ⊢ ¬ 𝐹 ∈ V
17		reseq2 5929	. . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On))
18	4	tfr1a 8323	. . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
19	18	simpli 483	. . . . . . . . 9 ⊢ Fun 𝐹
20		funrel 6503	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝐹
22	18	simpri 485	. . . . . . . . 9 ⊢ Lim dom 𝐹
23		limord 6372	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
24		ordsson 7723	. . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
25	22, 23, 24	mp2b 10	. . . . . . . 8 ⊢ dom 𝐹 ⊆ On
26		relssres 5977	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
27	21, 25, 26	mp2an 692	. . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹
28	17, 27	eqtrdi 2780	. . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹)
29	28	eleq1d 2813	. . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V))
30	16, 29	mtbiri 327	. . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V)
31	14, 30	2falsed 376	. . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
32	9, 31	jaoi 857	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
33	1, 32	sylbi 217	1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  dom cdm 5623   ↾ cres 5625  Rel wrel 5628  Ord word 6310  Oncon0 6311  Lim wlim 6312  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  recscrecs 8300

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-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301

This theorem is referenced by:  ordtypelem3  9431  ordtypelem9  9437