Theorem ofc12 6254

Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
ofc12.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofc12.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofc12.3	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
ofc12	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofc12

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofc12.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		ofc12.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
4		ofc12.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	4	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
6		fconstmpt 4771	. . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	a1i 9	. . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8		fconstmpt 4771	. . . 4 ⊢ (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶)
9	8	a1i 9	. . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶))
10	1, 3, 5, 7, 9	offval2 6246	. 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
11		fconstmpt 4771	. 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))
12	10, 11	eqtr4di 2280	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  {csn 3667   ↦ cmpt 4148   × cxp 4721  (class class class)co 6013   ∘_𝑓 cof 6228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by: (None)