Theorem caofrss 5958

Description: Transfer a relation subset 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)
caofcom.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofrss.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))

Assertion

Ref	Expression
caofrss	⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caofrss

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2	1	ffvelrnda 5507	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelrnda 5507	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofrss.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
6	5	ralrimivva 2486	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
7	6	adantr 272	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
8		breq1 3896	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
9		breq1 3896	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦 ↔ (𝐹‘𝑤)𝑇𝑦))
10	8, 9	imbi12d 233	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 → 𝑥𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦)))
11		breq2 3897	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		breq2 3897	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦 ↔ (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
13	11, 12	imbi12d 233	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤))))
14	10, 13	rspc2va 2771	. . . 4 ⊢ ((((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦)) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
15	2, 4, 7, 14	syl21anc 1196	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
16	15	ralimdva 2471	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
17		ffn 5228	. . . 4 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
18	1, 17	syl 14	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
19		ffn 5228	. . . 4 ⊢ (𝐺:𝐴⟶𝑆 → 𝐺 Fn 𝐴)
20	3, 19	syl 14	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
21		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
22		inidm 3249	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
23		eqidd 2114	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
24		eqidd 2114	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
25	18, 20, 21, 21, 22, 23, 24	ofrfval 5942	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
26	18, 20, 21, 21, 22, 23, 24	ofrfval 5942	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑇𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
27	16, 25, 26	3imtr4d 202	1 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1312   ∈ wcel 1461  ∀wral 2388   class class class wbr 3893   Fn wfn 5074  ⟶wf 5075  ‘cfv 5079   ∘_𝑟 cofr 5933

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ofr 5935

This theorem is referenced by: (None)