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Mirrors > Home > ILE Home > Th. List > mpteq2dv | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
mpteq2dv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq2dv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 272 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | mpteq2dva 3976 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 ↦ cmpt 3947 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-ral 2393 df-opab 3948 df-mpt 3949 |
This theorem is referenced by: ofeq 5936 rdgeq1 6219 rdgeq2 6220 omv 6302 oeiv 6303 0tonninf 10098 1tonninf 10099 iseqf1olemjpcl 10154 iseqf1olemqpcl 10155 iseqf1olemfvp 10156 seq3f1olemqsum 10159 seq3f1olemp 10161 summodc 11037 zsumdc 11038 fsum3 11041 sloteq 11800 cnprcl2k 12210 fsumcncntop 12535 expcncf 12571 peano4nninf 12877 peano3nninf 12878 nninfalllem1 12880 nninfsellemdc 12883 nninfsellemeq 12887 nninfsellemqall 12888 nninfsellemeqinf 12889 nninfomni 12892 |
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