Theorem caoftrn 7713

Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofcom.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofass.4	⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
caoftrn.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))

Assertion

Ref	Expression
caoftrn	⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) → 𝐹 ∘r 𝑈𝐻))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥,𝑈,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem caoftrn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caoftrn.5	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
2	1	ralrimivvva 3217	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
3	2	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4		caofref.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
5	4	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		breq1 5113	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
11	10	anbi1d 642	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧)))
12		breq1 5113	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑈𝑧 ↔ (𝐹‘𝑤)𝑈𝑧))
13	11, 12	imbi12d 347	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
14		breq2 5114	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15		breq1 5113	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑇𝑧 ↔ (𝐺‘𝑤)𝑇𝑧))
16	14, 15	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧)))
17	16	imbi1d 344	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
18		breq2 5114	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑇𝑧 ↔ (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
19	18	anbi2d 641	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
20		breq2 5114	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑈𝑧 ↔ (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
21	19, 20	imbi12d 347	. . . . . 6 ⊢ (𝑧 = (𝐻‘𝑤) → ((((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤))))
22	13, 17, 21	rspc3v 3606	. . . . 5 ⊢ (((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆 ∧ (𝐻‘𝑤) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤))))
23	5, 7, 9, 22	syl3anc 1396	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤))))
24	3, 23	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
25	24	ralimdva 3183	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
26	4	ffnd 6704	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	6	ffnd 6704	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
28		caofref.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
29		inidm 4187	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
30		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
31		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
32	26, 27, 28, 28, 29, 30, 31	ofrfval 7682	. . . 4 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
33	8	ffnd 6704	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐴)
34		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
35	27, 33, 28, 28, 29, 31, 34	ofrfval 7682	. . . 4 ⊢ (𝜑 → (𝐺 ∘r 𝑇𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
36	32, 35	anbi12d 643	. . 3 ⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) ↔ (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
37		r19.26 3131	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) ↔ (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
38	36, 37	bitr4di 292	. 2 ⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
39	26, 33, 28, 28, 29, 30, 34	ofrfval 7682	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑈𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
40	25, 38, 39	3imtr4d 297	1 ⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) → 𝐹 ∘r 𝑈𝐻))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5110  ⟶wf 6529  ‘cfv 6533   ∘_r cofr 7671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ofr 7673

This theorem is referenced by:  gsumbagdiaglem  22046  psdmul  22294  itg2le  25863