Theorem caoftrn 7692

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 3203	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
3	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4		caofref.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
5	4	ffvelcdmda 7072	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelcdmda 7072	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelcdmda 7072	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		breq1 5145	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
11	10	anbi1d 630	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧)))
12		breq1 5145	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑈𝑧 ↔ (𝐹‘𝑤)𝑈𝑧))
13	11, 12	imbi12d 344	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
14		breq2 5146	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15		breq1 5145	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑇𝑧 ↔ (𝐺‘𝑤)𝑇𝑧))
16	14, 15	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧)))
17	16	imbi1d 341	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
18		breq2 5146	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑇𝑧 ↔ (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
19	18	anbi2d 629	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
20		breq2 5146	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑈𝑧 ↔ (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
21	19, 20	imbi12d 344	. . . . . 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 3624	. . . . 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 1371	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
25	24	ralimdva 3167	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
26	4	ffnd 6706	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	6	ffnd 6706	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
28		caofref.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
29		inidm 4215	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
30		eqidd 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
31		eqidd 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
32	26, 27, 28, 28, 29, 30, 31	ofrfval 7664	. . . 4 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
33	8	ffnd 6706	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐴)
34		eqidd 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
35	27, 33, 28, 28, 29, 31, 34	ofrfval 7664	. . . 4 ⊢ (𝜑 → (𝐺 ∘r 𝑇𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
36	32, 35	anbi12d 631	. . 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 3111	. . 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 288	. 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 7664	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑈𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
40	25, 38, 39	3imtr4d 293	1 ⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) → 𝐹 ∘r 𝑈𝐻))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5142  ⟶wf 6529  ‘cfv 6533   ∘_r cofr 7653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ofr 7655

This theorem is referenced by:  gsumbagdiaglemOLD  21424  gsumbagdiaglem  21427  itg2le  25188