![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvmpt3 | Structured version Visualization version 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 2822 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) |
3 | fvmpt3.c | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) | |
4 | 2, 3 | vtoclga 3532 | . 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉) |
5 | fvmpt3.b | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 1, 5 | fvmptg 6943 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
7 | 4, 6 | mpdan 685 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5186 ‘cfv 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 |
This theorem is referenced by: fvmpt3i 6950 harval 9492 mrcfval 17480 elmptrab 23162 frlmsnic 40689 wallispi 44243 1arymaptfv 46658 2arymaptfv 46669 |
Copyright terms: Public domain | W3C validator |