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Mirrors > Home > ILE Home > Th. List > mpteq2dva | GIF version |
Description: Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) |
Ref | Expression |
---|---|
mpteq2dva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2dva | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | mpteq2dva.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
3 | 1, 2 | mpteq2da 3987 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ↦ cmpt 3959 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-opab 3960 df-mpt 3961 |
This theorem is referenced by: mpteq2dv 3989 fmptapd 5579 offval 5957 offval2 5965 caofinvl 5972 caofcom 5973 freceq1 6257 freceq2 6258 mapxpen 6710 xpmapenlem 6711 fser0const 10257 sumeq1 11092 sumeq2 11096 restid2 12056 cnmpt1t 12381 cnmpt12 12383 fsumcncntop 12652 divccncfap 12673 cdivcncfap 12683 expcncf 12688 dvidlemap 12756 dvcnp2cntop 12759 dvaddxxbr 12761 dvmulxxbr 12762 dvimulf 12766 dvcoapbr 12767 dvcjbr 12768 dvcj 12769 dvfre 12770 dvexp 12771 dvexp2 12772 dvrecap 12773 dvmptcmulcn 12779 dvmptnegcn 12780 dvmptsubcn 12781 dvef 12783 |
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