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Theorem caoftrn 5678
Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (φA 𝑉)
caofref.2 (φ𝐹:A𝑆)
caofcom.3 (φ𝐺:A𝑆)
caofass.4 (φ𝐻:A𝑆)
caoftrn.5 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝑅y y𝑇z) → x𝑈z))
Assertion
Ref Expression
caoftrn (φ → ((𝐹𝑟 𝑅𝐺 𝐺𝑟 𝑇𝐻) → 𝐹𝑟 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.
StepHypRef Expression
1 caoftrn.5 . . . . . 6 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝑅y y𝑇z) → x𝑈z))
21ralrimivvva 2396 . . . . 5 (φx 𝑆 y 𝑆 z 𝑆 ((x𝑅y y𝑇z) → x𝑈z))
32adantr 261 . . . 4 ((φ w A) → x 𝑆 y 𝑆 z 𝑆 ((x𝑅y y𝑇z) → x𝑈z))
4 caofref.2 . . . . . 6 (φ𝐹:A𝑆)
54ffvelrnda 5245 . . . . 5 ((φ w A) → (𝐹w) 𝑆)
6 caofcom.3 . . . . . 6 (φ𝐺:A𝑆)
76ffvelrnda 5245 . . . . 5 ((φ w A) → (𝐺w) 𝑆)
8 caofass.4 . . . . . 6 (φ𝐻:A𝑆)
98ffvelrnda 5245 . . . . 5 ((φ w A) → (𝐻w) 𝑆)
10 breq1 3758 . . . . . . . 8 (x = (𝐹w) → (x𝑅y ↔ (𝐹w)𝑅y))
1110anbi1d 438 . . . . . . 7 (x = (𝐹w) → ((x𝑅y y𝑇z) ↔ ((𝐹w)𝑅y y𝑇z)))
12 breq1 3758 . . . . . . 7 (x = (𝐹w) → (x𝑈z ↔ (𝐹w)𝑈z))
1311, 12imbi12d 223 . . . . . 6 (x = (𝐹w) → (((x𝑅y y𝑇z) → x𝑈z) ↔ (((𝐹w)𝑅y y𝑇z) → (𝐹w)𝑈z)))
14 breq2 3759 . . . . . . . 8 (y = (𝐺w) → ((𝐹w)𝑅y ↔ (𝐹w)𝑅(𝐺w)))
15 breq1 3758 . . . . . . . 8 (y = (𝐺w) → (y𝑇z ↔ (𝐺w)𝑇z))
1614, 15anbi12d 442 . . . . . . 7 (y = (𝐺w) → (((𝐹w)𝑅y y𝑇z) ↔ ((𝐹w)𝑅(𝐺w) (𝐺w)𝑇z)))
1716imbi1d 220 . . . . . 6 (y = (𝐺w) → ((((𝐹w)𝑅y y𝑇z) → (𝐹w)𝑈z) ↔ (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇z) → (𝐹w)𝑈z)))
18 breq2 3759 . . . . . . . 8 (z = (𝐻w) → ((𝐺w)𝑇z ↔ (𝐺w)𝑇(𝐻w)))
1918anbi2d 437 . . . . . . 7 (z = (𝐻w) → (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇z) ↔ ((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w))))
20 breq2 3759 . . . . . . 7 (z = (𝐻w) → ((𝐹w)𝑈z ↔ (𝐹w)𝑈(𝐻w)))
2119, 20imbi12d 223 . . . . . 6 (z = (𝐻w) → ((((𝐹w)𝑅(𝐺w) (𝐺w)𝑇z) → (𝐹w)𝑈z) ↔ (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) → (𝐹w)𝑈(𝐻w))))
2213, 17, 21rspc3v 2659 . . . . 5 (((𝐹w) 𝑆 (𝐺w) 𝑆 (𝐻w) 𝑆) → (x 𝑆 y 𝑆 z 𝑆 ((x𝑅y y𝑇z) → x𝑈z) → (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) → (𝐹w)𝑈(𝐻w))))
235, 7, 9, 22syl3anc 1134 . . . 4 ((φ w A) → (x 𝑆 y 𝑆 z 𝑆 ((x𝑅y y𝑇z) → x𝑈z) → (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) → (𝐹w)𝑈(𝐻w))))
243, 23mpd 13 . . 3 ((φ w A) → (((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) → (𝐹w)𝑈(𝐻w)))
2524ralimdva 2381 . 2 (φ → (w A ((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) → w A (𝐹w)𝑈(𝐻w)))
26 ffn 4989 . . . . . 6 (𝐹:A𝑆𝐹 Fn A)
274, 26syl 14 . . . . 5 (φ𝐹 Fn A)
28 ffn 4989 . . . . . 6 (𝐺:A𝑆𝐺 Fn A)
296, 28syl 14 . . . . 5 (φ𝐺 Fn A)
30 caofref.1 . . . . 5 (φA 𝑉)
31 inidm 3140 . . . . 5 (AA) = A
32 eqidd 2038 . . . . 5 ((φ w A) → (𝐹w) = (𝐹w))
33 eqidd 2038 . . . . 5 ((φ w A) → (𝐺w) = (𝐺w))
3427, 29, 30, 30, 31, 32, 33ofrfval 5662 . . . 4 (φ → (𝐹𝑟 𝑅𝐺w A (𝐹w)𝑅(𝐺w)))
35 ffn 4989 . . . . . 6 (𝐻:A𝑆𝐻 Fn A)
368, 35syl 14 . . . . 5 (φ𝐻 Fn A)
37 eqidd 2038 . . . . 5 ((φ w A) → (𝐻w) = (𝐻w))
3829, 36, 30, 30, 31, 33, 37ofrfval 5662 . . . 4 (φ → (𝐺𝑟 𝑇𝐻w A (𝐺w)𝑇(𝐻w)))
3934, 38anbi12d 442 . . 3 (φ → ((𝐹𝑟 𝑅𝐺 𝐺𝑟 𝑇𝐻) ↔ (w A (𝐹w)𝑅(𝐺w) w A (𝐺w)𝑇(𝐻w))))
40 r19.26 2435 . . 3 (w A ((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w)) ↔ (w A (𝐹w)𝑅(𝐺w) w A (𝐺w)𝑇(𝐻w)))
4139, 40syl6bbr 187 . 2 (φ → ((𝐹𝑟 𝑅𝐺 𝐺𝑟 𝑇𝐻) ↔ w A ((𝐹w)𝑅(𝐺w) (𝐺w)𝑇(𝐻w))))
4227, 36, 30, 30, 31, 32, 37ofrfval 5662 . 2 (φ → (𝐹𝑟 𝑈𝐻w A (𝐹w)𝑈(𝐻w)))
4325, 41, 423imtr4d 192 1 (φ → ((𝐹𝑟 𝑅𝐺 𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
Colors of variables: wff set class
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-bnd 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)
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