Theorem tfri1dALT 6503

Description: Alternate proof of tfri1d 6487 in terms of tfr1on 6502.

Although this does show that the tfr1on 6502 proof is general enough to also prove tfri1d 6487, the tfri1d 6487 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 6472	. . . 4 ⊢ Fun recs(𝐺)
2		tfri1dALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
3	2	funeqi 5339	. . . 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 6470	. . . . 5 ⊢ Ord dom recs(𝐺)
8	2	dmeqi 4924	. . . . . 6 ⊢ dom 𝐹 = dom recs(𝐺)
9		ordeq 4463	. . . . . 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 4584	. . . 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 4578	. . . . . . . 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 5629	. . . . . . . . . . . 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 4593	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
31		unon 4603	. . . . . . . . 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 4593	. . . . . . . 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 6502	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37		vex 2802	. . . . . . 7 ⊢ 𝑧 ∈ V
38	37	sucid 4508	. . . . . 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 5321	. 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 3888  Ord word 4453  Oncon0 4454  suc csuc 4456  dom cdm 4719   ↾ cres 4721  Fun wfun 5312   Fn wfn 5313  ‘cfv 5318  recscrecs 6456

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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

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 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-recs 6457

This theorem is referenced by: (None)