Theorem caofrss 7569

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 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofrss.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
6	5	ralrimivva 3123	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
8		breq1 5077	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
9		breq1 5077	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦 ↔ (𝐹‘𝑤)𝑇𝑦))
10	8, 9	imbi12d 345	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 → 𝑥𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦)))
11		breq2 5078	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		breq2 5078	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦 ↔ (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
13	11, 12	imbi12d 345	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤))))
14	10, 13	rspc2va 3571	. . . 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 835	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
16	15	ralimdva 3108	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
17	1	ffnd 6601	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	3	ffnd 6601	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
19		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		inidm 4152	. . 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 7543	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
24	17, 18, 19, 19, 20, 21, 22	ofrfval 7543	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑇𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
25	16, 23, 24	3imtr4d 294	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 → 𝐹 ∘r 𝑇𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433   ∘_r cofr 7532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ofr 7534

This theorem is referenced by: (None)