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Theorem caoftrn 7462
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 3104 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
32adantr 484 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6861 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6861 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6861 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 breq1 5033 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
1110anbi1d 633 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧)))
12 breq1 5033 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑈𝑧 ↔ (𝐹𝑤)𝑈𝑧))
1311, 12imbi12d 348 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) ↔ (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
14 breq2 5034 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
15 breq1 5033 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑇𝑧 ↔ (𝐺𝑤)𝑇𝑧))
1614, 15anbi12d 634 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧)))
1716imbi1d 345 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦𝑦𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧)))
18 breq2 5034 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑇𝑧 ↔ (𝐺𝑤)𝑇(𝐻𝑤)))
1918anbi2d 632 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
20 breq2 5034 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑈𝑧 ↔ (𝐹𝑤)𝑈(𝐻𝑤)))
2119, 20imbi12d 348 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇𝑧) → (𝐹𝑤)𝑈𝑧) ↔ (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
2213, 17, 21rspc3v 3539 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧) → (((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → (𝐹𝑤)𝑈(𝐻𝑤))))
235, 7, 9, 22syl3anc 1372 . . . 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 3091 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) → ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
264ffnd 6505 . . . . 5 (𝜑𝐹 Fn 𝐴)
276ffnd 6505 . . . . 5 (𝜑𝐺 Fn 𝐴)
28 caofref.1 . . . . 5 (𝜑𝐴𝑉)
29 inidm 4109 . . . . 5 (𝐴𝐴) = 𝐴
30 eqidd 2739 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
31 eqidd 2739 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
3226, 27, 28, 28, 29, 30, 31ofrfval 7434 . . . 4 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
338ffnd 6505 . . . . 5 (𝜑𝐻 Fn 𝐴)
34 eqidd 2739 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
3527, 33, 28, 28, 29, 31, 34ofrfval 7434 . . . 4 (𝜑 → (𝐺r 𝑇𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
3632, 35anbi12d 634 . . 3 (𝜑 → ((𝐹r 𝑅𝐺𝐺r 𝑇𝐻) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤))))
37 r19.26 3084 . . 3 (∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤)) ↔ (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) ∧ ∀𝑤𝐴 (𝐺𝑤)𝑇(𝐻𝑤)))
3836, 37bitr4di 292 . 2 (𝜑 → ((𝐹r 𝑅𝐺𝐺r 𝑇𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑤)𝑅(𝐺𝑤) ∧ (𝐺𝑤)𝑇(𝐻𝑤))))
3926, 33, 28, 28, 29, 30, 34ofrfval 7434 . 2 (𝜑 → (𝐹r 𝑈𝐻 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑈(𝐻𝑤)))
4025, 38, 393imtr4d 297 1 (𝜑 → ((𝐹r 𝑅𝐺𝐺r 𝑇𝐻) → 𝐹r 𝑈𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053   class class class wbr 5030  wf 6335  cfv 6339  r cofr 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ofr 7426
This theorem is referenced by:  gsumbagdiaglemOLD  20751  gsumbagdiaglem  20754  itg2le  24492
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