Theorem tfri1dALT 6319

Description: Alternate proof of tfri1d 6303 in terms of tfr1on 6318.

Although this does show that the tfr1on 6318 proof is general enough to also prove tfri1d 6303, the tfri1d 6303 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 6288	. . . 4 ⊢ Fun recs(𝐺)
2		tfri1dALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
3	2	funeqi 5209	. . . 4 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
4	1, 3	mpbir 145	. . 3 ⊢ Fun 𝐹
5	4	a1i 9	. 2 ⊢ (𝜑 → Fun 𝐹)
6		eqid 2165	. . . . . 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 6286	. . . . 5 ⊢ Ord dom recs(𝐺)
8	2	dmeqi 4805	. . . . . 6 ⊢ dom 𝐹 = dom recs(𝐺)
9		ordeq 4350	. . . . . 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 145	. . . 4 ⊢ Ord dom 𝐹
12		ordsson 4469	. . . 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 108	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → Fun 𝐺)
16	15	alimi 1443	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥Fun 𝐺)
17	14, 16	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥Fun 𝐺)
18	17	19.21bi 1546	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐺)
19	18	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → Fun 𝐺)
20		ordon 4463	. . . . . . . 8 ⊢ Ord On
21	20	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → Ord On)
22		simpr 109	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → (𝐺‘𝑥) ∈ V)
23	22	alimi 1443	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥(𝐺‘𝑥) ∈ V)
24		fveq2 5486	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝐺‘𝑥) = (𝐺‘𝑓))
25	24	eleq1d 2235	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝐺‘𝑥) ∈ V ↔ (𝐺‘𝑓) ∈ V))
26	25	spv 1848	. . . . . . . . . 10 ⊢ (∀𝑥(𝐺‘𝑥) ∈ V → (𝐺‘𝑓) ∈ V)
27	14, 23, 26	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑓) ∈ V)
28	27	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐺‘𝑓) ∈ V)
29	28	3ad2ant1 1008	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺‘𝑓) ∈ V)
30		suceloni 4478	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
31		unon 4488	. . . . . . . . 9 ⊢ ∪ On = On
32	30, 31	eleq2s 2261	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ On → suc 𝑦 ∈ On)
33	32	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
34		suceloni 4478	. . . . . . . 8 ⊢ (𝑧 ∈ On → suc 𝑧 ∈ On)
35	34	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ∈ On)
36	2, 19, 21, 29, 33, 35	tfr1on 6318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37		vex 2729	. . . . . . 7 ⊢ 𝑧 ∈ V
38	37	sucid 4395	. . . . . 6 ⊢ 𝑧 ∈ suc 𝑧
39		ssel2 3137	. . . . . 6 ⊢ ((suc 𝑧 ⊆ dom 𝐹 ∧ 𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹)
40	36, 38, 39	sylancl 410	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ dom 𝐹)
41	40	ex 114	. . . 4 ⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹))
42	41	ssrdv 3148	. . 3 ⊢ (𝜑 → On ⊆ dom 𝐹)
43	13, 42	eqssd 3159	. 2 ⊢ (𝜑 → dom 𝐹 = On)
44		df-fn 5191	. 2 ⊢ (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On))
45	5, 43, 44	sylanbrc 414	1 ⊢ (𝜑 → 𝐹 Fn On)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789  Ord word 4340  Oncon0 4341  suc csuc 4343  dom cdm 4604   ↾ cres 4606  Fun wfun 5182   Fn wfn 5183  ‘cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273

This theorem is referenced by: (None)