Theorem caofrss 7547

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 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelrnda 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofrss.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
6	5	ralrimivva 3114	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
8		breq1 5073	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
9		breq1 5073	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦 ↔ (𝐹‘𝑤)𝑇𝑦))
10	8, 9	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 → 𝑥𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦)))
11		breq2 5074	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		breq2 5074	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦 ↔ (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
13	11, 12	imbi12d 344	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤))))
14	10, 13	rspc2va 3563	. . . 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 834	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
16	15	ralimdva 3102	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
17	1	ffnd 6585	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	3	ffnd 6585	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
19		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		inidm 4149	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		eqidd 2739	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
22		eqidd 2739	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
23	17, 18, 19, 19, 20, 21, 22	ofrfval 7521	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
24	17, 18, 19, 19, 20, 21, 22	ofrfval 7521	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑇𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
25	16, 23, 24	3imtr4d 293	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 → 𝐹 ∘r 𝑇𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418   ∘_r cofr 7510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ofr 7512

This theorem is referenced by: (None)