Theorem mpteq1d 4174

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
mpteq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
mpteq1d	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem mpteq1d

Step	Hyp	Ref	Expression
1		mpteq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		mpteq1 4173	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1397   ↦ cmpt 4150

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-opab 4151  df-mpt 4152

This theorem is referenced by:  mptimass  5089  fmptapd  5845  offval  6243  swrd00g  11234  swrdlend  11243  swrd0g  11245  qusex  13413  mulgnn0gsum  13720  gsumfzconst  13933  gsumfzsnfd  13937  gsumfzfsumlem0  14606  gsumfzfsumlemm  14607  gsumgfsumlem  16709  gsumgfsum  16710