Theorem tfri1dALT 6256

Description: Alternate proof of tfri1d 6240 in terms of tfr1on 6255.

Although this does show that the tfr1on 6255 proof is general enough to also prove tfri1d 6240, the tfri1d 6240 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 6225	. . . 4 ⊢ Fun recs(𝐺)
2		tfri1dALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
3	2	funeqi 5152	. . . 4 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
4	1, 3	mpbir 145	. . 3 ⊢ Fun 𝐹
5	4	a1i 9	. 2 ⊢ (𝜑 → Fun 𝐹)
6		eqid 2140	. . . . . 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 6223	. . . . 5 ⊢ Ord dom recs(𝐺)
8	2	dmeqi 4748	. . . . . 6 ⊢ dom 𝐹 = dom recs(𝐺)
9		ordeq 4302	. . . . . 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 4416	. . . 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 1432	. . . . . . . . . 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 4410	. . . . . . . 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 1432	. . . . . . . . . 10 ⊢ (∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥(𝐺‘𝑥) ∈ V)
24		fveq2 5429	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝐺‘𝑥) = (𝐺‘𝑓))
25	24	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝐺‘𝑥) ∈ V ↔ (𝐺‘𝑓) ∈ V))
26	25	spv 1833	. . . . . . . . . 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 4425	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
31		unon 4435	. . . . . . . . 9 ⊢ ∪ On = On
32	30, 31	eleq2s 2235	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ On → suc 𝑦 ∈ On)
33	32	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
34		suceloni 4425	. . . . . . . 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 6255	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37		vex 2692	. . . . . . 7 ⊢ 𝑧 ∈ V
38	37	sucid 4347	. . . . . 6 ⊢ 𝑧 ∈ suc 𝑧
39		ssel2 3097	. . . . . 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 3108	. . 3 ⊢ (𝜑 → On ⊆ dom 𝐹)
43	13, 42	eqssd 3119	. 2 ⊢ (𝜑 → dom 𝐹 = On)
44		df-fn 5134	. 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 1330   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ⊆ wss 3076  ∪ cuni 3744  Ord word 4292  Oncon0 4293  suc csuc 4295  dom cdm 4547   ↾ cres 4549  Fun wfun 5125   Fn wfn 5126  ‘cfv 5131  recscrecs 6209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210

This theorem is referenced by: (None)