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Theorem caoftrn 5763
 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 2419 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
32adantr 265 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 5329 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 5329 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 5329 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 breq1 3794 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
1110anbi1d 446 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧)))
12 breq1 3794 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑈𝑧 ↔ (𝐹𝑤)𝑈𝑧))
1311, 12imbi12d 227 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
14 breq2 3795 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
15 breq1 3794 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑇𝑧 ↔ (𝐺𝑤)𝑇𝑧))
1614, 15anbi12d 450 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧)))
1716imbi1d 224 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
18 breq2 3795 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑇𝑧 ↔ (𝐺𝑤)𝑇(𝐻𝑤)))
1918anbi2d 445 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
20 breq2 3795 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑈𝑧 ↔ (𝐹𝑤)𝑈(𝐻𝑤)))
2119, 20imbi12d 227 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
2213, 17, 21rspc3v 2687 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
235, 7, 9, 22syl3anc 1146 . . . 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 2404 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
26 ffn 5073 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
274, 26syl 14 . . . . 5 (𝜑𝐹 Fn 𝐴)
28 ffn 5073 . . . . . 6 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
296, 28syl 14 . . . . 5 (𝜑𝐺 Fn 𝐴)
30 caofref.1 . . . . 5 (𝜑𝐴𝑉)
31 inidm 3173 . . . . 5 (𝐴𝐴) = 𝐴
32 eqidd 2057 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
33 eqidd 2057 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
3427, 29, 30, 30, 31, 32, 33ofrfval 5747 . . . 4 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
35 ffn 5073 . . . . . 6 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
368, 35syl 14 . . . . 5 (𝜑𝐻 Fn 𝐴)
37 eqidd 2057 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
3829, 36, 30, 30, 31, 33, 37ofrfval 5747 . . . 4 (𝜑 → (𝐺𝑟 𝑇𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
3934, 38anbi12d 450 . . 3 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤))))
40 r19.26 2458 . . 3 (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
4139, 40syl6bbr 191 . 2 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
4227, 36, 30, 30, 31, 32, 37ofrfval 5747 . 2 (𝜑 → (𝐹𝑟 𝑈𝐻 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
4325, 41, 423imtr4d 196 1 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∀wral 2323   class class class wbr 3791   Fn wfn 4924  ⟶wf 4925  ‘cfv 4929   ∘𝑟 cofr 5738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ofr 5740 This theorem is referenced by: (None)
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