Theorem mpteq12dva 5211

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2142, ax-12 2178. (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 2747	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
3	2	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
4		mpteq12dv.1	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐶)
5	4	eleq2d 2821	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
6	5	anbi1d 631	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
7	3, 6	bitrd 279	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
8	7	opabbidv 5190	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
9		df-mpt 5207	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10		df-mpt 5207	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
11	8, 9, 10	3eqtr4g 2796	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {copab 5186   ↦ cmpt 5206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-opab 5187  df-mpt 5207

This theorem is referenced by:  mpteq12dv  5212  mpteq2dva  5219  pfxmpt  14701  reps  14793  repswccat  14809  cidpropd  17727  monpropd  17755  fucpropd  17998  curfpropd  18250  hofpropd  18284  yonffthlem  18299  ofco2  22394  pmatcollpw3fi1lem1  22729  rrxnm  25348  ushgredgedg  29213  ushgredgedgloop  29215  cshw1s2  32941  gsumpart  33056  gsumhashmul  33060  gsumwrd2dccat  33066  cycpm2tr  33135  sgnsv  33176  extdg1id  33712  ofcfval  34134  ccatmulgnn0dir  34579  signstf0  34605  curunc  37631  cncfiooicc  45890  dvcosax  45922  fourierdlem74  46176  fourierdlem75  46177  fourierdlem93  46195  smfsupxr  46812  smflimsuplem8  46823