Theorem caofrss 6267

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	ffvelcdmda 5782	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 5782	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofrss.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
6	5	ralrimivva 2614	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
8		breq1 4091	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
9		breq1 4091	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦 ↔ (𝐹‘𝑤)𝑇𝑦))
10	8, 9	imbi12d 234	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 → 𝑥𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦)))
11		breq2 4092	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		breq2 4092	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦 ↔ (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
13	11, 12	imbi12d 234	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤))))
14	10, 13	rspc2va 2924	. . . 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 1272	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
16	15	ralimdva 2599	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
17		ffn 5482	. . . 4 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
18	1, 17	syl 14	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
19		ffn 5482	. . . 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 3416	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
23		eqidd 2232	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
24		eqidd 2232	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
25	18, 20, 21, 21, 22, 23, 24	ofrfval 6244	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
26	18, 20, 21, 21, 22, 23, 24	ofrfval 6244	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑇𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
27	16, 25, 26	3imtr4d 203	1 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326   ∘_𝑟 cofr 6234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ofr 6236

This theorem is referenced by: (None)