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Theorem caofref 5643
Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (φA 𝑉)
caofref.2 (φ𝐹:A𝑆)
caofref.3 ((φ x 𝑆) → 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 5215 . . . 4 ((φ w A) → (𝐹w) 𝑆)
3 caofref.3 . . . . . 6 ((φ x 𝑆) → x𝑅x)
43ralrimiva 2361 . . . . 5 (φx 𝑆 x𝑅x)
54adantr 261 . . . 4 ((φ w A) → x 𝑆 x𝑅x)
6 id 19 . . . . . 6 (x = (𝐹w) → x = (𝐹w))
76, 6breq12d 3740 . . . . 5 (x = (𝐹w) → (x𝑅x ↔ (𝐹w)𝑅(𝐹w)))
87rspcv 2620 . . . 4 ((𝐹w) 𝑆 → (x 𝑆 x𝑅x → (𝐹w)𝑅(𝐹w)))
92, 5, 8sylc 56 . . 3 ((φ w A) → (𝐹w)𝑅(𝐹w))
109ralrimiva 2361 . 2 (φw A (𝐹w)𝑅(𝐹w))
11 ffn 4960 . . . 4 (𝐹:A𝑆𝐹 Fn A)
121, 11syl 14 . . 3 (φ𝐹 Fn A)
13 caofref.1 . . 3 (φA 𝑉)
14 inidm 3114 . . 3 (AA) = A
15 eqidd 2014 . . 3 ((φ w A) → (𝐹w) = (𝐹w))
1612, 12, 13, 13, 14, 15, 15ofrfval 5631 . 2 (φ → (𝐹𝑟 𝑅𝐹w A (𝐹w)𝑅(𝐹w)))
1710, 16mpbird 156 1 (φ𝐹𝑟 𝑅𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223   wcel 1366  wral 2275   class class class wbr 3727   Fn wfn 4812  wf 4813  cfv 4817  𝑟 cofr 5622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ofr 5624
This theorem is referenced by: (None)
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