Theorem rtrclind 15048

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 15045	. 2 ⊢ (𝜂 → (t*‘𝑅) = (t*rec‘𝑅))
3	1	dfrtrclrec2 15041	. . . . . 6 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 ↔ ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋))
4	3	biimpac 477	. . . . 5 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋)
5		simprl 769	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝜂)
6		simprrr 780	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝑛 ∈ ℕ0)
7		simprrl 779	. . . . . . . . . 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 15047	. . . . . . . . . 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 466	. . . . . . . 8 ⊢ (((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝜏)
19	18	expcom 412	. . . . . . 7 ⊢ ((𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0) → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏))
20	19	expcom 412	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏)))
21	20	rexlimiv 3137	. . . . 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 412	. . 3 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 → 𝜏))
24		breq 5151	. . . 4 ⊢ ((t*‘𝑅) = (t*rec‘𝑅) → (𝑆(t*‘𝑅)𝑋 ↔ 𝑆(t*rec‘𝑅)𝑋))
25	24	imbi1d 340	. . 3 ⊢ ((t*‘𝑅) = (t*rec‘𝑅) → ((𝑆(t*‘𝑅)𝑋 → 𝜏) ↔ (𝑆(t*rec‘𝑅)𝑋 → 𝜏)))
26	23, 25	imbitrrid 245	. 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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059   class class class wbr 5149  Rel wrel 5683  ‘cfv 6549  (class class class)co 7419  ℕ₀cn0 12505  t*crtcl 14969  ↑_𝑟crelexp 15002  t*reccrtrcl 15038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-seq 14003  df-rtrcl 14971  df-relexp 15003  df-rtrclrec 15039

This theorem is referenced by: (None)