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Mirrors > Home > ILE Home > Th. List > fvmptmap | GIF version |
Description: Special case of fvmpt 5466 for operator theorems. (Contributed by NM, 27-Nov-2007.) |
Ref | Expression |
---|---|
fvmptmap.1 | ⊢ 𝐶 ∈ V |
fvmptmap.2 | ⊢ 𝐷 ∈ V |
fvmptmap.3 | ⊢ 𝑅 ∈ V |
fvmptmap.4 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptmap.5 | ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑𝑚 𝐷) ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptmap | ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptmap.3 | . . 3 ⊢ 𝑅 ∈ V | |
2 | fvmptmap.2 | . . 3 ⊢ 𝐷 ∈ V | |
3 | 1, 2 | elmap 6539 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝐷) ↔ 𝐴:𝐷⟶𝑅) |
4 | fvmptmap.4 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | fvmptmap.5 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑𝑚 𝐷) ↦ 𝐵) | |
6 | fvmptmap.1 | . . 3 ⊢ 𝐶 ∈ V | |
7 | 4, 5, 6 | fvmpt 5466 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝐷) → (𝐹‘𝐴) = 𝐶) |
8 | 3, 7 | sylbir 134 | 1 ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ↦ cmpt 3959 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 ↑𝑚 cmap 6510 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-map 6512 |
This theorem is referenced by: (None) |
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