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Theorem caofref 7682
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 5154 . . . 4 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 481 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
76ffvelcdmda 7071 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
82, 5, 7rspcdva 3610 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
98ralrimiva 3145 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
106ffnd 6705 . . 3 (𝜑𝐹 Fn 𝐴)
11 caofref.1 . . 3 (𝜑𝐴𝑉)
12 inidm 4214 . . 3 (𝐴𝐴) = 𝐴
13 eqidd 2732 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1410, 10, 11, 11, 12, 13, 13ofrfval 7663 . 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 3060   class class class wbr 5141  wf 6528  cfv 6532  r cofr 7652
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ofr 7654
This theorem is referenced by:  psrridm  21455  itg2itg1  25183  itg20  25184
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