Theorem rtrclind 14990

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 14987	. 2 ⊢ (𝜂 → (t*‘𝑅) = (t*rec‘𝑅))
3	1	dfrtrclrec2 14983	. . . . . 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 14989	. . . . . . . . . 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 3123	. . . . 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 5097	. . . 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 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5095  Rel wrel 5628  ‘cfv 6486  (class class class)co 7353  ℕ₀cn0 12402  t*crtcl 14911  ↑_𝑟crelexp 14944  t*reccrtrcl 14980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-seq 13927  df-rtrcl 14913  df-relexp 14945  df-rtrclrec 14981

This theorem is referenced by: (None)