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Theorem caofref 6821
 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 6267 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 480 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 id 22 . . . . . 6 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
76, 6breq12d 4596 . . . . 5 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
87rspcv 3278 . . . 4 ((𝐹𝑤) ∈ 𝑆 → (∀𝑥𝑆 𝑥𝑅𝑥 → (𝐹𝑤)𝑅(𝐹𝑤)))
92, 5, 8sylc 63 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
109ralrimiva 2949 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
11 ffn 5958 . . . 4 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
121, 11syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
13 caofref.1 . . 3 (𝜑𝐴𝑉)
14 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
15 eqidd 2611 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1612, 12, 13, 13, 14, 15, 15ofrfval 6803 . 2 (𝜑 → (𝐹𝑟 𝑅𝐹 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤)))
1710, 16mpbird 246 1 (𝜑𝐹𝑟 𝑅𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804   ∘𝑟 cofr 6794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ofr 6796 This theorem is referenced by:  psrridm  19225  itg2itg1  23309  itg20  23310
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