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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 274 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | mpteq2dva 4066 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ↦ cmpt 4037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-ral 2447 df-opab 4038 df-mpt 4039 |
This theorem is referenced by: ofeq 6046 rdgeq1 6330 rdgeq2 6331 omv 6414 oeiv 6415 0tonninf 10364 1tonninf 10365 iseqf1olemjpcl 10420 iseqf1olemqpcl 10421 iseqf1olemfvp 10422 seq3f1olemqsum 10425 seq3f1olemp 10427 summodc 11310 zsumdc 11311 fsum3 11314 prodeq2w 11483 prodmodc 11505 zproddc 11506 fprodseq 11510 sloteq 12342 cnprcl2k 12753 fsumcncntop 13103 expcncf 13139 dvexp 13222 dvexp2 13223 peano4nninf 13727 peano3nninf 13728 nninfalllem1 13729 nninfsellemdc 13731 nninfsellemeq 13735 nninfsellemqall 13736 nninfsellemeqinf 13737 nninfomni 13740 |
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