Theorem caofrss 7721

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 7076	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
3		caofcom.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7076	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofrss.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
6	5	ralrimivva 3206	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦 → 𝑥𝑇𝑦))
8		breq1 5106	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑤)𝑅𝑦))
9		breq1 5106	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦 ↔ (𝐹‘𝑤)𝑇𝑦))
10	8, 9	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦 → 𝑥𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦)))
11		breq2 5107	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦 ↔ (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		breq2 5107	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦 ↔ (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
13	11, 12	imbi12d 347	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦 → (𝐹‘𝑤)𝑇𝑦) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤))))
14	10, 13	rspc2va 3588	. . . 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 851	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤) → (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
16	15	ralimdva 3175	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤) → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
17	1	ffnd 6702	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	3	ffnd 6702	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
19		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		inidm 4172	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
22		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
23	17, 18, 19, 19, 20, 21, 22	ofrfval 7692	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐺‘𝑤)))
24	17, 18, 19, 19, 20, 21, 22	ofrfval 7692	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑇𝐺 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑇(𝐺‘𝑤)))
25	16, 23, 24	3imtr4d 297	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 → 𝐹 ∘r 𝑇𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  ⟶wf 6527  ‘cfv 6531   ∘_r cofr 7681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ofr 7683

This theorem is used by: (None)