Theorem caofref 7707

Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofref.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥)

Assertion

Ref	Expression
caofref	⊢ (𝜑 → 𝐹 ∘r 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.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
2	1, 1	breq12d 5137	. . . 4 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑥 ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑤)))
3		caofref.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥)
4	3	ralrimiva 3133	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥)
6		caofref.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
7	6	ffvelcdmda 7079	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
8	2, 5, 7	rspcdva 3607	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤)𝑅(𝐹‘𝑤))
9	8	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤))
10	6	ffnd 6712	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		inidm 4207	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
13		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
14	10, 10, 11, 11, 12, 13, 13	ofrfval 7686	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐹 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤)))
15	9, 14	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052   class class class wbr 5124  ⟶wf 6532  ‘cfv 6536   ∘_r cofr 7675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ofr 7677

This theorem is referenced by:  psrridm  21928  itg2itg1  25694  itg20  25695