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Mirrors > Home > MPE Home > Th. List > mpteq12dva | Structured version Visualization version GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2135, ax-12 2169. (Revised by SN, 11-Nov-2024.) |
Ref | Expression |
---|---|
mpteq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12dva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12dva | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12dva.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
2 | 1 | eqeq2d 2741 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷)) |
3 | 2 | pm5.32da 577 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷))) |
4 | mpteq12dv.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐶) | |
5 | 4 | eleq2d 2817 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
6 | 5 | anbi1d 628 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
7 | 3, 6 | bitrd 278 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
8 | 7 | opabbidv 5213 | . 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}) |
9 | df-mpt 5231 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
10 | df-mpt 5231 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)} | |
11 | 8, 9, 10 | 3eqtr4g 2795 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {copab 5209 ↦ cmpt 5230 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-opab 5210 df-mpt 5231 |
This theorem is referenced by: mpteq12dv 5238 mpteq2dva 5247 pfxmpt 14632 reps 14724 repswccat 14740 cidpropd 17658 monpropd 17688 fucpropd 17934 curfpropd 18190 hofpropd 18224 yonffthlem 18239 ofco2 22173 pmatcollpw3fi1lem1 22508 rrxnm 25139 ushgredgedg 28753 ushgredgedgloop 28755 cshw1s2 32391 gsumpart 32477 gsumhashmul 32478 cycpm2tr 32548 sgnsv 32589 extdg1id 33030 ofcfval 33394 ccatmulgnn0dir 33851 signstf0 33877 curunc 36773 cncfiooicc 44908 dvcosax 44940 fourierdlem74 45194 fourierdlem75 45195 fourierdlem93 45213 smfsupxr 45830 smflimsuplem8 45841 |
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