Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caofrss GIF version

 Description: Transfer a relation subset 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)
caofrss.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
Assertion
Ref Expression
caofrss (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
21ffvelrnda 5330 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 5330 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofrss.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
65ralrimivva 2418 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
76adantr 265 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
8 breq1 3795 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
9 breq1 3795 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑇𝑦 ↔ (𝐹𝑤)𝑇𝑦))
108, 9imbi12d 227 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑥𝑇𝑦) ↔ ((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦)))
11 breq2 3796 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
12 breq2 3796 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑇𝑦 ↔ (𝐹𝑤)𝑇(𝐺𝑤)))
1311, 12imbi12d 227 . . . . 5 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤))))
1410, 13rspc2va 2686 . . . 4 ((((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦)) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
152, 4, 7, 14syl21anc 1145 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
1615ralimdva 2404 . 2 (𝜑 → (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) → ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
17 ffn 5074 . . . 4 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
181, 17syl 14 . . 3 (𝜑𝐹 Fn 𝐴)
19 ffn 5074 . . . 4 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
203, 19syl 14 . . 3 (𝜑𝐺 Fn 𝐴)
21 caofref.1 . . 3 (𝜑𝐴𝑉)
22 inidm 3174 . . 3 (𝐴𝐴) = 𝐴
23 eqidd 2057 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
24 eqidd 2057 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
2518, 20, 21, 21, 22, 23, 24ofrfval 5748 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
2618, 20, 21, 21, 22, 23, 24ofrfval 5748 . 2 (𝜑 → (𝐹𝑟 𝑇𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
2716, 25, 263imtr4d 196 1 (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  ∀wral 2323   class class class wbr 3792   Fn wfn 4925  ⟶wf 4926  ‘cfv 4930   ∘𝑟 cofr 5739 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 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972 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 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ofr 5741 This theorem is referenced by: (None)
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