Theorem ovmpot 7574

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 7422	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦) = (𝐴𝐹𝐵))
2		eqid 2760	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))
3		ovex 7446	. 2 ⊢ (𝐴𝐹𝐵) ∈ V
4	1, 2, 3	ovmpoa 7568	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  (class class class)co 7413   ∈ cmpo 7415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418

This theorem is used by:  cncrng  21606  cnfld1  21610  cndrng  21614  cnflddiv  21615  cnsubrglem  21630  expcn  25100  negcncf  25150  dvcnp2  26147  dvmulbr  26166  dvcobr  26173  cmvth  26218  dvfsumle  26248  dvfsumlem2  26254  dvply2g  26515  taylply2  26604  taylthlem2  26610  mpodvdsmulf1o  27430  fsumdvdsmul  27431