Theorem caoftrn 6086

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 2553	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
3	2	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4		caofref.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
5	4	ffvelrnda 5631	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelrnda 5631	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelrnda 5631	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		breq1 3992	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
11	10	anbi1d 462	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧)))
12		breq1 3992	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑈𝑧 ↔ (𝐹‘𝑤)𝑈𝑧))
13	11, 12	imbi12d 233	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
14		breq2 3993	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15		breq1 3992	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑇𝑧 ↔ (𝐺‘𝑤)𝑇𝑧))
16	14, 15	anbi12d 470	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧)))
17	16	imbi1d 230	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((((𝐹‘𝑤)𝑅𝑦 ∧ 𝑦𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) → (𝐹‘𝑤)𝑈𝑧)))
18		breq2 3993	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑇𝑧 ↔ (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
19	18	anbi2d 461	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇𝑧) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
20		breq2 3993	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑈𝑧 ↔ (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
21	19, 20	imbi12d 233	. . . . . 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 2850	. . . . 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 1233	. . . 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 13	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
25	24	ralimdva 2537	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤)) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
26		ffn 5347	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
27	4, 26	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
28		ffn 5347	. . . . . 6 ⊢ (𝐺:𝐴⟶𝑆 → 𝐺 Fn 𝐴)
29	6, 28	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
30		caofref.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
31		inidm 3336	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
32		eqidd 2171	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
33		eqidd 2171	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
34	27, 29, 30, 30, 31, 32, 33	ofrfval 6069	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
35		ffn 5347	. . . . . 6 ⊢ (𝐻:𝐴⟶𝑆 → 𝐻 Fn 𝐴)
36	8, 35	syl 14	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐴)
37		eqidd 2171	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
38	29, 36, 30, 30, 31, 33, 37	ofrfval 6069	. . . 4 ⊢ (𝜑 → (𝐺 ∘𝑟 𝑇𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤)))
39	34, 38	anbi12d 470	. . 3 ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) ↔ (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
40		r19.26 2596	. . 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)))
41	39, 40	bitr4di 197	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹‘𝑤)𝑅(𝐺‘𝑤) ∧ (𝐺‘𝑤)𝑇(𝐻‘𝑤))))
42	27, 36, 30, 30, 31, 32, 37	ofrfval 6069	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑈𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑈(𝐻‘𝑤)))
43	25, 41, 42	3imtr4d 202	1 ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) → 𝐹 ∘𝑟 𝑈𝐻))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∀wral 2448   class class class wbr 3989   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198   ∘_𝑟 cofr 6060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ofr 6062

This theorem is referenced by: (None)