Theorem tfri1dALT 6301

Description: Alternate proof of tfri1d 6285 in terms of tfr1on 6300.

Although this does show that the tfr1on 6300 proof is general enough to also prove tfri1d 6285, the tfri1d 6285 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 6270	. . . 4 ⊢ Fun recs(𝐺)
2		tfri1dALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
3	2	funeqi 5194	. . . 4 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
4	1, 3	mpbir 145	. . 3 ⊢ Fun 𝐹
5	4	a1i 9	. 2 ⊢ (𝜑 → Fun 𝐹)
6		eqid 2157	. . . . . 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 6268	. . . . 5 ⊢ Ord dom recs(𝐺)
8	2	dmeqi 4790	. . . . . 6 ⊢ dom 𝐹 = dom recs(𝐺)
9		ordeq 4335	. . . . . 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 4454	. . . 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 1435	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥Fun 𝐺)
17	14, 16	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥Fun 𝐺)
18	17	19.21bi 1538	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐺)
19	18	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → Fun 𝐺)
20		ordon 4448	. . . . . . . 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 1435	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥(𝐺‘𝑥) ∈ V)
24		fveq2 5471	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝐺‘𝑥) = (𝐺‘𝑓))
25	24	eleq1d 2226	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝐺‘𝑥) ∈ V ↔ (𝐺‘𝑓) ∈ V))
26	25	spv 1840	. . . . . . . . . 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 1003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺‘𝑓) ∈ V)
30		suceloni 4463	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
31		unon 4473	. . . . . . . . 9 ⊢ ∪ On = On
32	30, 31	eleq2s 2252	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ On → suc 𝑦 ∈ On)
33	32	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
34		suceloni 4463	. . . . . . . 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 6300	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37		vex 2715	. . . . . . 7 ⊢ 𝑧 ∈ V
38	37	sucid 4380	. . . . . 6 ⊢ 𝑧 ∈ suc 𝑧
39		ssel2 3123	. . . . . 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 3134	. . 3 ⊢ (𝜑 → On ⊆ dom 𝐹)
43	13, 42	eqssd 3145	. 2 ⊢ (𝜑 → dom 𝐹 = On)
44		df-fn 5176	. 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 1333   = wceq 1335   ∈ wcel 2128  {cab 2143  ∀wral 2435  ∃wrex 2436  Vcvv 2712   ⊆ wss 3102  ∪ cuni 3774  Ord word 4325  Oncon0 4326  suc csuc 4328  dom cdm 4589   ↾ cres 4591  Fun wfun 5167   Fn wfn 5168  ‘cfv 5173  recscrecs 6254

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-recs 6255

This theorem is referenced by: (None)