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Theorem caofref 5760
Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofref.3 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
Assertion
Ref Expression
caofref (𝜑𝐹𝑟 𝑅𝐹)
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem caofref
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
21ffvelrnda 5330 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 2409 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 265 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 id 19 . . . . . 6 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
76, 6breq12d 3805 . . . . 5 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
87rspcv 2669 . . . 4 ((𝐹𝑤) ∈ 𝑆 → (∀𝑥𝑆 𝑥𝑅𝑥 → (𝐹𝑤)𝑅(𝐹𝑤)))
92, 5, 8sylc 60 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
109ralrimiva 2409 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
11 ffn 5074 . . . 4 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
121, 11syl 14 . . 3 (𝜑𝐹 Fn 𝐴)
13 caofref.1 . . 3 (𝜑𝐴𝑉)
14 inidm 3174 . . 3 (𝐴𝐴) = 𝐴
15 eqidd 2057 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1612, 12, 13, 13, 14, 15, 15ofrfval 5748 . 2 (𝜑 → (𝐹𝑟 𝑅𝐹 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤)))
1710, 16mpbird 160 1 (𝜑𝐹𝑟 𝑅𝐹)
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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