Theorem ofcf 34287

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 6673	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcf.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcf.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
5		eqidd 2738	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
6	2, 3, 4, 5	ofcfval 34282	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶)))
7	1	ffvelcdmda 7040	. . 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 3181	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12		ovrspc2v 7396	. . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈)
13	7, 8, 11, 12	syl21anc 838	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈)
14	6, 13	fmpt3d 7072	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∘_f/c cofc 34279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-ofc 34280

This theorem is referenced by:  signshf  34772