Theorem fconstmpo 6050

Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)

Assertion

Ref	Expression
fconstmpo	⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem fconstmpo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstmpt 4727	. 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2		eqidd 2207	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
3	2	mpompt 6047	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	1, 3	eqtri 2227	1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1373  {csn 3635  ⟨cop 3638   ↦ cmpt 4110   × cxp 4678   ∈ cmpo 5956

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-iun 3932  df-opab 4111  df-mpt 4112  df-xp 4686  df-rel 4687  df-oprab 5958  df-mpo 5959

This theorem is referenced by: (None)