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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 2810 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) |
3 | fvmpt3.c | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) | |
4 | 2, 3 | vtoclga 3556 | . 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉) |
5 | fvmpt3.b | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 1, 5 | fvmptg 7002 | . 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 1533 ∈ wcel 2098 ↦ cmpt 5232 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: fvmpt3i 7009 harval 9585 mrcfval 17591 elmptrab 23775 zringfrac 33369 frlmsnic 41908 wallispi 45596 1arymaptfv 47899 2arymaptfv 47910 |
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