Theorem ofcfval2 31039

Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfval2.2	⊢ (𝜑 → 𝐶 ∈ 𝑊)
ofcfval2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑋)
ofcfval2.4	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))

Assertion

Ref	Expression
ofcfval2	⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem ofcfval2

Step	Hyp	Ref	Expression
1		ofcfval2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑋)
2	1	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑋)
3		eqid 2771	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	fnmpt 6315	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	syl 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		ofcfval2.4	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
7	6	fneq1d 6276	. . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
8	5, 7	mpbird 249	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9		ofcfval2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		ofcfval2.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
11	6, 1	fvmpt2d 6605	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
12	8, 9, 10, 11	ofcfval 31033	1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3081   ↦ cmpt 5004   Fn wfn 6180  (class class class)co 6974  ∘_𝑓/𝑐cofc 31030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-ofc 31031

This theorem is referenced by:  coinflippv  31419  ofcs1  31492