Theorem ovmpot 7557

Description: The value of an operation is equal to the value of the same operation expressed in maps-to notation. (Contributed by GG, 16-Mar-2025.) (Revised by GG, 13-Apr-2025.)

Assertion

Ref	Expression
ovmpot	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))

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

Proof of Theorem ovmpot

Step	Hyp	Ref	Expression
1		oveq12 7405	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦) = (𝐴𝐹𝐵))
2		eqid 2762	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))
3		ovex 7429	. 2 ⊢ (𝐴𝐹𝐵) ∈ V
4	1, 2, 3	ovmpoa 7551	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  (class class class)co 7396   ∈ cmpo 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401

This theorem is referenced by:  cncrng  21442  cnfld1  21446  cndrng  21450  cnflddiv  21451  cnsubrglem  21466  expcn  24931  negcncf  24981  dvcnp2  25979  dvmulbr  25998  dvcobr  26005  cmvth  26050  dvfsumle  26080  dvfsumlem2  26086  dvply2g  26346  taylply2  26428  taylthlem2  26434  mpodvdsmulf1o  27255  fsumdvdsmul  27256