Theorem tfr2b 8371

Description: Without assuming ax-rep 5232, 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 7769	. 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 8367	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
5	4	dmeqi 5885	. . . . 5 ⊢ dom 𝐹 = dom recs(𝐺)
6	5	eleq2i 2857	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺))
7	4	reseq1i 5965	. . . . 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 7765	. . . . . 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 8365	. . . . . 6 ⊢ ¬ recs(𝐺) ∈ V
16	4, 15	eqneltri 2884	. . . . 5 ⊢ ¬ 𝐹 ∈ V
17		reseq2 5964	. . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On))
18	4	tfr1a 8369	. . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
19	18	simpli 488	. . . . . . . . 9 ⊢ Fun 𝐹
20		funrel 6542	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝐹
22	18	simpri 490	. . . . . . . . 9 ⊢ Lim dom 𝐹
23		limord 6411	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
24		ordsson 7770	. . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
25	22, 23, 24	mp2b 10	. . . . . . . 8 ⊢ dom 𝐹 ⊆ On
26		relssres 6012	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
27	21, 25, 26	mp2an 704	. . . . . . 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 870	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  dom cdm 5652   ↾ cres 5654  Rel wrel 5657  Ord word 6349  Oncon0 6350  Lim wlim 6351  Fun wfun 6519   Fn wfn 6520  ‘cfv 6525  recscrecs 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346

This theorem is referenced by:  ordtypelem3  9470  ordtypelem9  9476