Theorem ofcf 33717

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 6718	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcf.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcf.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
5		eqidd 2729	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
6	2, 3, 4, 5	ofcfval 33712	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶)))
7	1	ffvelcdmda 7089	. . 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 3196	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12		ovrspc2v 7441	. . 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 7121	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∀wral 3057  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415   ∘_f/c cofc 33709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-ofc 33710

This theorem is referenced by:  signshf  34215