Theorem mpteq12dva 5165

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2152, ax-12 2189. (Revised by SN, 11-Nov-2024.)

Hypotheses

Ref	Expression
mpteq12dv.1	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12dva.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12dva	⊢ (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 mpteq12dva

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq12dva.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
2	1	eqeq2d 2751	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
3	2	pm5.32da 584	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
4		mpteq12dv.1	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐶)
5	4	eleq2d 2826	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
6	5	anbi1d 637	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
7	3, 6	bitrd 280	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
8	7	opabbidv 5145	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
9		df-mpt 5161	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10		df-mpt 5161	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
11	8, 9, 10	3eqtr4g 2800	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {copab 5141   ↦ cmpt 5160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-opab 5142  df-mpt 5161

This theorem is referenced by:  mpteq12dv  5166  mpteq2dva  5172  pfxmpt  14639  reps  14730  repswccat  14746  cidpropd  17674  monpropd  17702  fucpropd  17945  curfpropd  18197  hofpropd  18231  yonffthlem  18246  ofco2  22441  pmatcollpw3fi1lem1  22776  rrxnm  25383  ushgredgedg  29323  ushgredgedgloop  29325  cshw1s2  33046  gsumpart  33151  gsumhashmul  33155  gsumwrd2dccat  33166  cycpm2tr  33207  sgnsv  33248  extdg1id  33857  ofcfval  34289  ccatmulgnn0dir  34733  signstf0  34759  curunc  37976  cncfiooicc  46344  dvcosax  46376  fourierdlem74  46630  fourierdlem75  46631  fourierdlem93  46649  smfsupxr  47266  smflimsuplem8  47277  lmdpropd  50154  cmdpropd  50155