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Theorem caofref 7418
 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 5044 . . . 4 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑥 ↔ (𝐹𝑤)𝑅(𝐹𝑤)))
3 caofref.3 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)
43ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑥𝑆 𝑥𝑅𝑥)
54adantr 484 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆 𝑥𝑅𝑥)
6 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
76ffvelrnda 6829 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
82, 5, 7rspcdva 3573 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤)𝑅(𝐹𝑤))
98ralrimiva 3149 . 2 (𝜑 → ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤))
106ffnd 6489 . . 3 (𝜑𝐹 Fn 𝐴)
11 caofref.1 . . 3 (𝜑𝐴𝑉)
12 inidm 4145 . . 3 (𝐴𝐴) = 𝐴
13 eqidd 2799 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
1410, 10, 11, 11, 12, 13, 13ofrfval 7399 . 2 (𝜑 → (𝐹r 𝑅𝐹 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐹𝑤)))
159, 14mpbird 260 1 (𝜑𝐹r 𝑅𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5031  ⟶wf 6321  ‘cfv 6325   ∘r cofr 7390 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ofr 7392 This theorem is referenced by:  psrridm  20648  itg2itg1  24350  itg20  24351
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