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Mirrors > Home > ILE Home > Th. List > fvmptg | GIF version |
Description: Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (x = A → B = 𝐶) |
fvmptg.2 | ⊢ 𝐹 = (x ∈ 𝐷 ↦ B) |
Ref | Expression |
---|---|
fvmptg | ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘A) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2037 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | fvmptg.1 | . . . 4 ⊢ (x = A → B = 𝐶) | |
3 | 2 | eqeq2d 2048 | . . 3 ⊢ (x = A → (y = B ↔ y = 𝐶)) |
4 | eqeq1 2043 | . . 3 ⊢ (y = 𝐶 → (y = 𝐶 ↔ 𝐶 = 𝐶)) | |
5 | moeq 2710 | . . . 4 ⊢ ∃*y y = B | |
6 | 5 | a1i 9 | . . 3 ⊢ (x ∈ 𝐷 → ∃*y y = B) |
7 | fvmptg.2 | . . . 4 ⊢ 𝐹 = (x ∈ 𝐷 ↦ B) | |
8 | df-mpt 3811 | . . . 4 ⊢ (x ∈ 𝐷 ↦ B) = {〈x, y〉 ∣ (x ∈ 𝐷 ∧ y = B)} | |
9 | 7, 8 | eqtri 2057 | . . 3 ⊢ 𝐹 = {〈x, y〉 ∣ (x ∈ 𝐷 ∧ y = B)} |
10 | 3, 4, 6, 9 | fvopab3ig 5189 | . 2 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐶 = 𝐶 → (𝐹‘A) = 𝐶)) |
11 | 1, 10 | mpi 15 | 1 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘A) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∃*wmo 1898 {copab 3808 ↦ cmpt 3809 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: fvmpt 5192 fvmpts 5193 fvmpt3 5194 fvmpt2 5197 f1mpt 5353 fnofval 5663 caofinvl 5675 1stvalg 5711 2ndvalg 5712 brtpos2 5807 frec0g 5922 frecsuclem3 5929 sucinc 5964 sucinc2 5965 omcl 5980 oeicl 5981 oav2 5982 omv2 5984 cjval 9073 reval 9077 imval 9078 absval 9210 |
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