Theorem ovmpot 7516

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 7364	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦) = (𝐴𝐹𝐵))
2		eqid 2733	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))
3		ovex 7388	. 2 ⊢ (𝐴𝐹𝐵) ∈ V
4	1, 2, 3	ovmpoa 7510	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  (class class class)co 7355   ∈ cmpo 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360

This theorem is referenced by:  cncrng  21334  cnfld1  21339  cndrng  21344  cnflddiv  21346  cnsubrglem  21362  expcn  24810  negcncf  24862  dvcnp2  25868  dvmulbr  25888  dvcobr  25896  cmvth  25942  dvfsumle  25973  dvfsumlem2  25980  dvply2g  26239  taylply2  26322  taylthlem2  26329  mpodvdsmulf1o  27151  fsumdvdsmul  27152