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Mirrors > Home > ILE Home > Th. List > fvmpt3 | GIF version |
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
fvmpt3.a | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmpt3.b | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
fvmpt3.c | ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmpt3 | ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmpt3.a | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | 1 | eleq1d 2186 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) |
3 | fvmpt3.c | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) | |
4 | 2, 3 | vtoclga 2726 | . 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉) |
5 | fvmpt3.b | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 1, 5 | fvmptg 5465 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
7 | 4, 6 | mpdan 417 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ↦ cmpt 3959 ‘cfv 5093 |
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-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-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 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 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-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: fvmpt3i 5469 frec2uzsucd 10142 |
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