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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 2040 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | fvmptg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | eqeq2d 2051 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
4 | eqeq1 2046 | . . 3 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐶 ↔ 𝐶 = 𝐶)) | |
5 | moeq 2716 | . . . 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 3820 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} | |
9 | 7, 8 | eqtri 2060 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} |
10 | 3, 4, 6, 9 | fvopab3ig 5246 | . 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 1243 ∈ wcel 1393 ∃*wmo 1901 {copab 3817 ↦ cmpt 3818 ‘cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 |
This theorem is referenced by: fvmpt 5249 fvmpts 5250 fvmpt3 5251 fvmpt2 5254 f1mpt 5410 fnofval 5721 caofinvl 5733 1stvalg 5769 2ndvalg 5770 brtpos2 5866 frec0g 5983 frecsuclem3 5990 sucinc 6025 sucinc2 6026 omcl 6041 oeicl 6042 oav2 6043 omv2 6045 cardval3ex 6365 ceilqval 9148 monoord2 9236 iseqdistr 9249 serile 9253 cjval 9445 reval 9449 imval 9450 cvg1nlemcau 9583 cvg1nlemres 9584 absval 9599 resqrexlemglsq 9620 resqrexlemga 9621 climmpt 9821 climle 9854 climcvg1nlem 9868 |
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