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Theorem caofref 7698
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 id 22 . . . . 5 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
21, 1breq12d 5161 . . . 4 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 481 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
76ffvelcdmda 7086 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
82, 5, 7rspcdva 3613 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
98ralrimiva 3146 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
106ffnd 6718 . . 3 (𝜑𝐹 Fn 𝐴)
11 caofref.1 . . 3 (𝜑𝐴𝑉)
12 inidm 4218 . . 3 (𝐴𝐴) = 𝐴
13 eqidd 2733 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1410, 10, 11, 11, 12, 13, 13ofrfval 7679 . 2 (𝜑 → (𝐹r 𝑅𝐹 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤)))
159, 14mpbird 256 1 (𝜑𝐹r 𝑅𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061   class class class wbr 5148  wf 6539  cfv 6543  r cofr 7668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ofr 7670
This theorem is referenced by:  psrridm  21523  itg2itg1  25253  itg20  25254
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