Theorem tfri1dALT 6487

Description: Alternate proof of tfri1d 6471 in terms of tfr1on 6486.

Although this does show that the tfr1on 6486 proof is general enough to also prove tfri1d 6471, the tfri1d 6471 proof is simpler in places because it does not need to deal with 𝑋 being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

Hypotheses

Ref	Expression
tfri1dALT.1	⊢ 𝐹 = recs(𝐺)
tfri1dALT.2	⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))

Assertion

Ref	Expression
tfri1dALT	⊢ (𝜑 → 𝐹 Fn On)

Distinct variable group:   𝑥,𝐺

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem tfri1dALT

Dummy variables 𝑧 𝑎 𝑏 𝑐 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrfun 6456	. . . 4 ⊢ Fun recs(𝐺)
2		tfri1dALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
3	2	funeqi 5335	. . . 4 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
4	1, 3	mpbir 146	. . 3 ⊢ Fun 𝐹
5	4	a1i 9	. 2 ⊢ (𝜑 → Fun 𝐹)
6		eqid 2229	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}
7	6	tfrlem8 6454	. . . . 5 ⊢ Ord dom recs(𝐺)
8	2	dmeqi 4921	. . . . . 6 ⊢ dom 𝐹 = dom recs(𝐺)
9		ordeq 4460	. . . . . 6 ⊢ (dom 𝐹 = dom recs(𝐺) → (Ord dom 𝐹 ↔ Ord dom recs(𝐺)))
10	8, 9	ax-mp 5	. . . . 5 ⊢ (Ord dom 𝐹 ↔ Ord dom recs(𝐺))
11	7, 10	mpbir 146	. . . 4 ⊢ Ord dom 𝐹
12		ordsson 4581	. . . 4 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
13	11, 12	mp1i 10	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ On)
14		tfri1dALT.2	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
15		simpl 109	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → Fun 𝐺)
16	15	alimi 1501	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥Fun 𝐺)
17	14, 16	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥Fun 𝐺)
18	17	19.21bi 1604	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐺)
19	18	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → Fun 𝐺)
20		ordon 4575	. . . . . . . 8 ⊢ Ord On
21	20	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → Ord On)
22		simpr 110	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → (𝐺‘𝑥) ∈ V)
23	22	alimi 1501	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥(𝐺‘𝑥) ∈ V)
24		fveq2 5623	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝐺‘𝑥) = (𝐺‘𝑓))
25	24	eleq1d 2298	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝐺‘𝑥) ∈ V ↔ (𝐺‘𝑓) ∈ V))
26	25	spv 1906	. . . . . . . . . 10 ⊢ (∀𝑥(𝐺‘𝑥) ∈ V → (𝐺‘𝑓) ∈ V)
27	14, 23, 26	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑓) ∈ V)
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐺‘𝑓) ∈ V)
29	28	3ad2ant1 1042	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺‘𝑓) ∈ V)
30		onsuc 4590	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
31		unon 4600	. . . . . . . . 9 ⊢ ∪ On = On
32	30, 31	eleq2s 2324	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ On → suc 𝑦 ∈ On)
33	32	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
34		onsuc 4590	. . . . . . . 8 ⊢ (𝑧 ∈ On → suc 𝑧 ∈ On)
35	34	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ∈ On)
36	2, 19, 21, 29, 33, 35	tfr1on 6486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37		vex 2802	. . . . . . 7 ⊢ 𝑧 ∈ V
38	37	sucid 4505	. . . . . 6 ⊢ 𝑧 ∈ suc 𝑧
39		ssel2 3219	. . . . . 6 ⊢ ((suc 𝑧 ⊆ dom 𝐹 ∧ 𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹)
40	36, 38, 39	sylancl 413	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ dom 𝐹)
41	40	ex 115	. . . 4 ⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹))
42	41	ssrdv 3230	. . 3 ⊢ (𝜑 → On ⊆ dom 𝐹)
43	13, 42	eqssd 3241	. 2 ⊢ (𝜑 → dom 𝐹 = On)
44		df-fn 5317	. 2 ⊢ (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On))
45	5, 43, 44	sylanbrc 417	1 ⊢ (𝜑 → 𝐹 Fn On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ⊆ wss 3197  ∪ cuni 3887  Ord word 4450  Oncon0 4451  suc csuc 4453  dom cdm 4716   ↾ cres 4718  Fun wfun 5308   Fn wfn 5309  ‘cfv 5314  recscrecs 6440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-recs 6441

This theorem is referenced by: (None)