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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 276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | mpteq2dva 4108 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ↦ cmpt 4079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-ral 2473 df-opab 4080 df-mpt 4081 |
This theorem is referenced by: ofeqd 6109 ofeq 6110 rdgeq1 6397 rdgeq2 6398 omv 6481 oeiv 6482 0tonninf 10472 1tonninf 10473 iseqf1olemjpcl 10528 iseqf1olemqpcl 10529 iseqf1olemfvp 10530 seq3f1olemqsum 10533 seq3f1olemp 10535 summodc 11426 zsumdc 11427 fsum3 11430 prodeq2w 11599 prodmodc 11621 zproddc 11622 fprodseq 11626 1arithlem1 12398 sloteq 12520 qusex 12805 grplactfval 13060 cnprcl2k 14183 fsumcncntop 14533 expcncf 14569 dvexp 14652 dvexp2 14653 lgsval 14883 peano4nninf 15234 peano3nninf 15235 nninfalllem1 15236 nninfsellemdc 15238 nninfsellemeq 15242 nninfsellemqall 15243 nninfsellemeqinf 15244 nninfomni 15247 |
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