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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 2897 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) |
3 | fvmpt3.c | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) | |
4 | 2, 3 | vtoclga 3574 | . 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉) |
5 | fvmpt3.b | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 1, 5 | fvmptg 6766 | . 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 1537 ∈ wcel 2114 ↦ cmpt 5146 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: fvmpt3i 6773 harval 9026 mrcfval 16879 elmptrab 22435 frlmsnic 39169 wallispi 42375 |
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