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Theorem caofref 7718
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 5163 . . . 4 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 3142 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 479 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
76ffvelcdmda 7097 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
82, 5, 7rspcdva 3610 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
98ralrimiva 3142 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
106ffnd 6726 . . 3 (𝜑𝐹 Fn 𝐴)
11 caofref.1 . . 3 (𝜑𝐴𝑉)
12 inidm 4219 . . 3 (𝐴𝐴) = 𝐴
13 eqidd 2728 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1410, 10, 11, 11, 12, 13, 13ofrfval 7699 . 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 394   = wceq 1533  wcel 2098  wral 3057   class class class wbr 5150  wf 6547  cfv 6551  r cofr 7688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ofr 7690
This theorem is referenced by:  psrridm  21911  itg2itg1  25684  itg20  25685
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