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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.) |
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 | mpteq12dv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | 1 | alrimiv 1924 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
3 | mpteq12dva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
4 | 3 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
5 | mpteq12f 5141 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
6 | 2, 4, 5 | syl2anc 586 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ↦ cmpt 5138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-opab 5121 df-mpt 5139 |
This theorem is referenced by: mpteq12dvOLD 5144 pfxmpt 14034 reps 14126 repswccat 14142 cidpropd 16974 monpropd 17001 fucpropd 17241 curfpropd 17477 hofpropd 17511 yonffthlem 17526 ofco2 21054 pmatcollpw3fi1lem1 21388 rrxnm 23988 ushgredgedg 27005 ushgredgedgloop 27007 cshw1s2 30629 cycpm2tr 30756 sgnsv 30797 extdg1id 31048 ofcfval 31352 ccatmulgnn0dir 31807 signstf0 31833 curunc 34868 cncfiooicc 42170 dvcosax 42204 fourierdlem74 42459 fourierdlem75 42460 fourierdlem93 42478 smfsupxr 43084 smflimsuplem8 43095 |
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