Theorem rtrclind 14867

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 14864	. 2 ⊢ (𝜂 → (t*‘𝑅) = (t*rec‘𝑅))
3	1	dfrtrclrec2 14860	. . . . . 6 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 ↔ ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋))
4	3	biimpac 479	. . . . 5 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → ∃𝑛 ∈ ℕ0 𝑆(𝑅↑𝑟𝑛)𝑋)
5		simprl 768	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝜂)
6		simprrr 779	. . . . . . . . . 10 ⊢ ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0))) → 𝑛 ∈ ℕ0)
7		simprrl 778	. . . . . . . . . 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 14866	. . . . . . . . . 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 468	. . . . . . . 8 ⊢ (((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) ∧ (𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝜏)
19	18	expcom 414	. . . . . . 7 ⊢ ((𝑆(𝑅↑𝑟𝑛)𝑋 ∧ 𝑛 ∈ ℕ0) → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏))
20	19	expcom 414	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋 ∧ 𝜂) → 𝜏)))
21	20	rexlimiv 3141	. . . . 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 414	. . 3 ⊢ (𝜂 → (𝑆(t*rec‘𝑅)𝑋 → 𝜏))
24		breq 5091	. . . 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	syl5ibr 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 396   = wceq 1540   ∈ wcel 2105  ∃wrex 3070   class class class wbr 5089  Rel wrel 5619  ‘cfv 6473  (class class class)co 7329  ℕ₀cn0 12326  t*crtcl 14788  ↑_𝑟crelexp 14821  t*reccrtrcl 14857

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-n0 12327  df-z 12413  df-uz 12676  df-seq 13815  df-rtrcl 14790  df-relexp 14822  df-rtrclrec 14858

This theorem is referenced by: (None)