Theorem rtrclind 14969

Description: Principle of transitive induction. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)

Hypotheses

Ref	Expression
rtrclind.1	⊢ (𝜂 → Rel 𝑅)
rtrclind.2	⊢ (𝜂 → 𝑆 ∈ 𝑉)
rtrclind.3	⊢ (𝜂 → 𝑋 ∈ 𝑊)
rtrclind.4	⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))
rtrclind.5	⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))
rtrclind.6	⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))
rtrclind.7	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏))
rtrclind.8	⊢ (𝜂 → 𝜒)
rtrclind.9	⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))

Assertion

Ref	Expression
rtrclind	⊢ (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏))

Distinct variable groups:   𝑥,𝑅,𝑖,𝑗   𝑥,𝑆,𝑖,𝑗   𝑥,𝑋   𝜂,𝑥,𝑖,𝑗   𝜏,𝑥   𝜓,𝑖,𝑗   𝜃,𝑖   𝜑,𝑗,𝑥   𝜒,𝑖

Allowed substitution hints:   𝜑(𝑖)   𝜓(𝑥)   𝜒(𝑥,𝑗)   𝜃(𝑥,𝑗)   𝜏(𝑖,𝑗)   𝑉(𝑥,𝑖,𝑗)   𝑊(𝑥,𝑖,𝑗)   𝑋(𝑖,𝑗)

Proof of Theorem rtrclind

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rtrclind.1	. . 3 ⊢ (𝜂 → Rel 𝑅)
2	1	dfrtrcl2 14966	. 2 ⊢ (𝜂 → (t*‘𝑅) = (t*rec‘𝑅))
3	1	dfrtrclrec2 14962	. . . . . 6 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 ↔ ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋))
4	3	biimpac 478	. . . . 5 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋)
5		simprl 770	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝜂)
6		simprrr 781	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝑛 ∈ ℕ0)
7		simprrl 780	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝑆(𝑅↑𝑟𝑛)𝑋)
8		rtrclind.2	. . . . . . . . . . 11 ⊢ (𝜂 → 𝑆 ∈ 𝑉)
9		rtrclind.3	. . . . . . . . . . 11 ⊢ (𝜂 → 𝑋 ∈ 𝑊)
10		rtrclind.4	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))
11		rtrclind.5	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))
12		rtrclind.6	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))
13		rtrclind.7	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏))
14		rtrclind.8	. . . . . . . . . . 11 ⊢ (𝜂 → 𝜒)
15		rtrclind.9	. . . . . . . . . . 11 ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))
16	1, 8, 9, 10, 11, 12, 13, 14, 15	relexpind 14968	. . . . . . . . . 10 ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))
17	5, 6, 7, 16	syl3c 66	. . . . . . . . 9 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝜏)
18	17	anassrs 467	. . . . . . . 8 ⊢ (((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝜏)
19	18	expcom 413	. . . . . . 7 ⊢ ((𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0) → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏))
20	19	expcom 413	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏)))
21	20	rexlimiv 3126	. . . . 5 ⊢ (∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏))
22	4, 21	mpcom 38	. . . 4 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏)
23	22	expcom 413	. . 3 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 → 𝜏))
24		breq 5093	. . . 4 ⊢ ((t*‘𝑅) = (t*rec‘𝑅) → (𝑆(t*‘𝑅)𝑋 ↔ 𝑆(t*rec‘𝑅)𝑋))
25	24	imbi1d 341	. . 3 ⊢ ((t*‘𝑅) = (t*rec‘𝑅) → ((𝑆(t*‘𝑅)𝑋 → 𝜏) ↔ (𝑆(t*rec‘𝑅)𝑋 → 𝜏)))
26	23, 25	imbitrrid 246	. 2 ⊢ ((t*‘𝑅) = (t*rec‘𝑅) → (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏)))
27	2, 26	mpcom 38	1 ⊢ (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   class class class wbr 5091  Rel wrel 5621  ‘cfv 6481  (class class class)co 7346  ℕ₀cn0 12378  t*crtcl 14890  ↑_𝑟crelexp 14923  t*reccrtrcl 14959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-n0 12379  df-z 12466  df-uz 12730  df-seq 13906  df-rtrcl 14892  df-relexp 14924  df-rtrclrec 14960

This theorem is referenced by: (None)