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Theorem caoftrn 5788
Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofcom.3 (𝜑𝐺:𝐴𝑆)
caofass.4 (𝜑𝐻:𝐴𝑆)
caoftrn.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
Assertion
Ref Expression
caoftrn (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 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 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
21ralrimivvva 2449 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
32adantr 270 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 5355 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 5355 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 5355 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 breq1 3808 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
1110anbi1d 453 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧)))
12 breq1 3808 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑈𝑧 ↔ (𝐹𝑤)𝑈𝑧))
1311, 12imbi12d 232 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
14 breq2 3809 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
15 breq1 3808 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑇𝑧 ↔ (𝐺𝑤)𝑇𝑧))
1614, 15anbi12d 457 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧)))
1716imbi1d 229 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
18 breq2 3809 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑇𝑧 ↔ (𝐺𝑤)𝑇(𝐻𝑤)))
1918anbi2d 452 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
20 breq2 3809 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑈𝑧 ↔ (𝐹𝑤)𝑈(𝐻𝑤)))
2119, 20imbi12d 232 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
2213, 17, 21rspc3v 2724 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
235, 7, 9, 22syl3anc 1170 . . . 4 ((𝜑𝑤𝐴) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
243, 23mpd 13 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤)))
2524ralimdva 2434 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
26 ffn 5097 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
274, 26syl 14 . . . . 5 (𝜑𝐹 Fn 𝐴)
28 ffn 5097 . . . . . 6 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
296, 28syl 14 . . . . 5 (𝜑𝐺 Fn 𝐴)
30 caofref.1 . . . . 5 (𝜑𝐴𝑉)
31 inidm 3191 . . . . 5 (𝐴𝐴) = 𝐴
32 eqidd 2084 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
33 eqidd 2084 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
3427, 29, 30, 30, 31, 32, 33ofrfval 5772 . . . 4 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
35 ffn 5097 . . . . . 6 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
368, 35syl 14 . . . . 5 (𝜑𝐻 Fn 𝐴)
37 eqidd 2084 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
3829, 36, 30, 30, 31, 33, 37ofrfval 5772 . . . 4 (𝜑 → (𝐺𝑟 𝑇𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
3934, 38anbi12d 457 . . 3 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤))))
40 r19.26 2490 . . 3 (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
4139, 40syl6bbr 196 . 2 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
4227, 36, 30, 30, 31, 32, 37ofrfval 5772 . 2 (𝜑 → (𝐹𝑟 𝑈𝐻 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
4325, 41, 423imtr4d 201 1 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wral 2353   class class class wbr 3805   Fn wfn 4947  wf 4948  cfv 4952  𝑟 cofr 5763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ofr 5765
This theorem is referenced by: (None)
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