Theorem mpt2eq3dv 6619

Description: An equality deduction for the maps to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)

Hypothesis

Ref	Expression
mpt2eq3dv.1	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
mpt2eq3dv	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem mpt2eq3dv

Step	Hyp	Ref	Expression
1		mpt2eq3dv.1	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
2	1	3ad2ant1 1075	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
3	2	mpt2eq3dva 6617	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  seqomeq12  7436  cantnfval  8448  seqeq2  12667  seqeq3  12668  relexpsucnnr  13613  lsmfval  17876  phssip  19822  mamuval  20011  matsc  20075  marrepval0  20186  marrepval  20187  marepvval0  20191  marepvval  20192  submaval0  20205  mdetr0  20230  mdet0  20231  mdetunilem7  20243  mdetunilem8  20244  madufval  20262  maduval  20263  maducoeval2  20265  madutpos  20267  madugsum  20268  madurid  20269  minmar1val0  20272  minmar1val  20273  pmat0opsc  20322  pmat1opsc  20323  mat2pmatval  20348  cpm2mval  20374  decpmatid  20394  pmatcollpw2lem  20401  pmatcollpw3lem  20407  mply1topmatval  20428  mp2pm2mplem1  20430  mp2pm2mplem4  20433  ttgval  25555  smatfval  29189  ofceq  29486  finxpeq1  32399  matunitlindflem1  32575  idfusubc  41656  digfval  42189