![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fvmptmap | GIF version |
Description: Special case of fvmpt 5506 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 6579 | . 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 5506 | . 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 1332 ∈ wcel 1481 Vcvv 2689 ↦ cmpt 3997 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ↑𝑚 cmap 6550 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |