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Theorem caoftrn 5837
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 2452 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
32adantr 270 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 5397 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 5397 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 5397 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 breq1 3823 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
1110anbi1d 453 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧)))
12 breq1 3823 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑈𝑧 ↔ (𝐹𝑤)𝑈𝑧))
1311, 12imbi12d 232 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
14 breq2 3824 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
15 breq1 3823 . . . . . . . 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 3824 . . . . . . . 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 3824 . . . . . . 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 2729 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
235, 7, 9, 22syl3anc 1172 . . . 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 2437 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
26 ffn 5126 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
274, 26syl 14 . . . . 5 (𝜑𝐹 Fn 𝐴)
28 ffn 5126 . . . . . 6 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
296, 28syl 14 . . . . 5 (𝜑𝐺 Fn 𝐴)
30 caofref.1 . . . . 5 (𝜑𝐴𝑉)
31 inidm 3198 . . . . 5 (𝐴𝐴) = 𝐴
32 eqidd 2086 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
33 eqidd 2086 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
3427, 29, 30, 30, 31, 32, 33ofrfval 5821 . . . 4 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
35 ffn 5126 . . . . . 6 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
368, 35syl 14 . . . . 5 (𝜑𝐻 Fn 𝐴)
37 eqidd 2086 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
3829, 36, 30, 30, 31, 33, 37ofrfval 5821 . . . 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 2493 . . 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 5821 . 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 922   = wceq 1287  wcel 1436  wral 2355   class class class wbr 3820   Fn wfn 4976  wf 4977  cfv 4981  𝑟 cofr 5812
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ofr 5814
This theorem is referenced by: (None)
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