Theorem caoftrn 5678

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 2396	. . . . 5 ⊢ (φ → ∀x ∈ 𝑆 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝑅y ∧ y𝑇z) → x𝑈z))
3	2	adantr 261	. . . 4 ⊢ ((φ ∧ w ∈ A) → ∀x ∈ 𝑆 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝑅y ∧ y𝑇z) → x𝑈z))
4		caofref.2	. . . . . 6 ⊢ (φ → 𝐹:A⟶𝑆)
5	4	ffvelrnda 5245	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐹‘w) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (φ → 𝐺:A⟶𝑆)
7	6	ffvelrnda 5245	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐺‘w) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (φ → 𝐻:A⟶𝑆)
9	8	ffvelrnda 5245	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐻‘w) ∈ 𝑆)
10		breq1 3758	. . . . . . . 8 ⊢ (x = (𝐹‘w) → (x𝑅y ↔ (𝐹‘w)𝑅y))
11	10	anbi1d 438	. . . . . . 7 ⊢ (x = (𝐹‘w) → ((x𝑅y ∧ y𝑇z) ↔ ((𝐹‘w)𝑅y ∧ y𝑇z)))
12		breq1 3758	. . . . . . 7 ⊢ (x = (𝐹‘w) → (x𝑈z ↔ (𝐹‘w)𝑈z))
13	11, 12	imbi12d 223	. . . . . 6 ⊢ (x = (𝐹‘w) → (((x𝑅y ∧ y𝑇z) → x𝑈z) ↔ (((𝐹‘w)𝑅y ∧ y𝑇z) → (𝐹‘w)𝑈z)))
14		breq2 3759	. . . . . . . 8 ⊢ (y = (𝐺‘w) → ((𝐹‘w)𝑅y ↔ (𝐹‘w)𝑅(𝐺‘w)))
15		breq1 3758	. . . . . . . 8 ⊢ (y = (𝐺‘w) → (y𝑇z ↔ (𝐺‘w)𝑇z))
16	14, 15	anbi12d 442	. . . . . . 7 ⊢ (y = (𝐺‘w) → (((𝐹‘w)𝑅y ∧ y𝑇z) ↔ ((𝐹‘w)𝑅(𝐺‘w) ∧ (𝐺‘w)𝑇z)))
17	16	imbi1d 220	. . . . . 6 ⊢ (y = (𝐺‘w) → ((((𝐹‘w)𝑅y ∧ y𝑇z) → (𝐹‘w)𝑈z) ↔ (((𝐹‘w)𝑅(𝐺‘w) ∧ (𝐺‘w)𝑇z) → (𝐹‘w)𝑈z)))
18		breq2 3759	. . . . . . . 8 ⊢ (z = (𝐻‘w) → ((𝐺‘w)𝑇z ↔ (𝐺‘w)𝑇(𝐻‘w)))
19	18	anbi2d 437	. . . . . . 7 ⊢ (z = (𝐻‘w) → (((𝐹‘w)𝑅(𝐺‘w) ∧ (𝐺‘w)𝑇z) ↔ ((𝐹‘w)𝑅(𝐺‘w) ∧ (𝐺‘w)𝑇(𝐻‘w))))
20		breq2 3759	. . . . . . 7 ⊢ (z = (𝐻‘w) → ((𝐹‘w)𝑈z ↔ (𝐹‘w)𝑈(𝐻‘w)))
21	19, 20	imbi12d 223	. . . . . 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 2659	. . . . 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 1134	. . . 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 2381	. 2 ⊢ (φ → (∀w ∈ A ((𝐹‘w)𝑅(𝐺‘w) ∧ (𝐺‘w)𝑇(𝐻‘w)) → ∀w ∈ A (𝐹‘w)𝑈(𝐻‘w)))
26		ffn 4989	. . . . . 6 ⊢ (𝐹:A⟶𝑆 → 𝐹 Fn A)
27	4, 26	syl 14	. . . . 5 ⊢ (φ → 𝐹 Fn A)
28		ffn 4989	. . . . . 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 3140	. . . . 5 ⊢ (A ∩ A) = A
32		eqidd 2038	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐹‘w) = (𝐹‘w))
33		eqidd 2038	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐺‘w) = (𝐺‘w))
34	27, 29, 30, 30, 31, 32, 33	ofrfval 5662	. . . 4 ⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀w ∈ A (𝐹‘w)𝑅(𝐺‘w)))
35		ffn 4989	. . . . . 6 ⊢ (𝐻:A⟶𝑆 → 𝐻 Fn A)
36	8, 35	syl 14	. . . . 5 ⊢ (φ → 𝐻 Fn A)
37		eqidd 2038	. . . . 5 ⊢ ((φ ∧ w ∈ A) → (𝐻‘w) = (𝐻‘w))
38	29, 36, 30, 30, 31, 33, 37	ofrfval 5662	. . . 4 ⊢ (φ → (𝐺 ∘𝑟 𝑇𝐻 ↔ ∀w ∈ A (𝐺‘w)𝑇(𝐻‘w)))
39	34, 38	anbi12d 442	. . 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 2435	. . 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	syl6bbr 187	. 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 5662	. 2 ⊢ (φ → (𝐹 ∘𝑟 𝑈𝐻 ↔ ∀w ∈ A (𝐹‘w)𝑈(𝐻‘w)))
43	25, 41, 42	3imtr4d 192	1 ⊢ (φ → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) → 𝐹 ∘𝑟 𝑈𝐻))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300   class class class wbr 3755   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845   ∘_𝑟 cofr 5653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ofr 5655

This theorem is referenced by: (None)