Theorem mpteq12dv 4166

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12dv.1	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12dv.2	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12dv	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem mpteq12dv

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		mpteq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
3	2	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
4	1, 3	mpteq12dva 4165	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200   ↦ cmpt 4145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-opab 4146  df-mpt 4147

This theorem is referenced by:  mpteq12i  4172  offval  6235  offval3  6288  ccatfvalfi  11145  swrdval  11201  odzval  12785  restval  13299  prdsex  13323  prdsval  13327  qusval  13377  grpinvfvalg  13596  grpinvpropdg  13629  opprnegg  14067  lspfval  14373  lsppropd  14417  sraval  14422  psrval  14651  ntrfval  14795  clsfval  14796  neifval  14835  cnpfval  14890  cnprcl2k  14901  reldvg  15374  dvfvalap  15376  eldvap  15377  vtxdgfval  16074  vtxdgop  16078  vtxdeqd  16082