Theorem ofcf 34267

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 6665	. . 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 34262	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶)))
7	1	ffvelcdmda 7032	. . 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 7388	. . 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 7064	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∘_f/c cofc 34259

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 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-ofc 34260

This theorem is referenced by:  signshf  34752