Theorem ofcf 34105

Description: The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
ofcf.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
ofcf.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
ofcf.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcf.5	⊢ (𝜑 → 𝐶 ∈ 𝑇)

Assertion

Ref	Expression
ofcf	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)

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

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

Proof of Theorem ofcf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcf.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2	1	ffnd 6736	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcf.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcf.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
5		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
6	2, 3, 4, 5	ofcfval 34100	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶)))
7	1	ffvelcdmda 7103	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆)
8	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑇)
9		ofcf.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
10	9	ralrimivva 3201	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12		ovrspc2v 7458	. . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈)
13	7, 8, 11, 12	syl21anc 837	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈)
14	6, 13	fmpt3d 7135	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  ∀wral 3060  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∘_f/c cofc 34097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-ofc 34098

This theorem is referenced by:  signshf  34604