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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 2826 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) | 
| 3 | fvmpt3.c | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) | |
| 4 | 2, 3 | vtoclga 3577 | . 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉) | 
| 5 | fvmpt3.b | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 6 | 1, 5 | fvmptg 7014 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) | 
| 7 | 4, 6 | mpdan 687 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: fvmpt3i 7021 harval 9600 mrcfval 17651 elmptrab 23835 zringfrac 33582 frlmsnic 42550 wallispi 46085 1arymaptfv 48561 2arymaptfv 48572 | 
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