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Theorem caoftrn 6897
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 2968 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
32adantr 481 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6325 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6325 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6325 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 breq1 4626 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
1110anbi1d 740 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧)))
12 breq1 4626 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑈𝑧 ↔ (𝐹𝑤)𝑈𝑧))
1311, 12imbi12d 334 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
14 breq2 4627 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
15 breq1 4626 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑇𝑧 ↔ (𝐺𝑤)𝑇𝑧))
1614, 15anbi12d 746 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧)))
1716imbi1d 331 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
18 breq2 4627 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑇𝑧 ↔ (𝐺𝑤)𝑇(𝐻𝑤)))
1918anbi2d 739 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
20 breq2 4627 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑈𝑧 ↔ (𝐹𝑤)𝑈(𝐻𝑤)))
2119, 20imbi12d 334 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
2213, 17, 21rspc3v 3314 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
235, 7, 9, 22syl3anc 1323 . . . 4 ((𝜑𝑤𝐴) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
243, 23mpd 15 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤)))
2524ralimdva 2958 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
26 ffn 6012 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
274, 26syl 17 . . . . 5 (𝜑𝐹 Fn 𝐴)
28 ffn 6012 . . . . . 6 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
296, 28syl 17 . . . . 5 (𝜑𝐺 Fn 𝐴)
30 caofref.1 . . . . 5 (𝜑𝐴𝑉)
31 inidm 3806 . . . . 5 (𝐴𝐴) = 𝐴
32 eqidd 2622 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
33 eqidd 2622 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
3427, 29, 30, 30, 31, 32, 33ofrfval 6870 . . . 4 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
35 ffn 6012 . . . . . 6 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
368, 35syl 17 . . . . 5 (𝜑𝐻 Fn 𝐴)
37 eqidd 2622 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
3829, 36, 30, 30, 31, 33, 37ofrfval 6870 . . . 4 (𝜑 → (𝐺𝑟 𝑇𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
3934, 38anbi12d 746 . . 3 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤))))
40 r19.26 3059 . . 3 (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
4139, 40syl6bbr 278 . 2 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
4227, 36, 30, 30, 31, 32, 37ofrfval 6870 . 2 (𝜑 → (𝐹𝑟 𝑈𝐻 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
4325, 41, 423imtr4d 283 1 (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623   Fn wfn 5852  wf 5853  cfv 5857  𝑟 cofr 6861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ofr 6863
This theorem is referenced by:  gsumbagdiaglem  19315  itg2le  23446
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