Theorem ovmpot 7572

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 7420	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦) = (𝐴𝐹𝐵))
2		eqid 2769	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))
3		ovex 7444	. 2 ⊢ (𝐴𝐹𝐵) ∈ V
4	1, 2, 3	ovmpoa 7566	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  (class class class)co 7411   ∈ cmpo 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  cncrng  21511  cnfld1  21515  cndrng  21519  cnflddiv  21520  cnsubrglem  21535  expcn  24999  negcncf  25049  dvcnp2  26047  dvmulbr  26066  dvcobr  26073  cmvth  26118  dvfsumle  26148  dvfsumlem2  26154  dvply2g  26414  taylply2  26496  taylthlem2  26502  mpodvdsmulf1o  27323  fsumdvdsmul  27324