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Mirrors > Home > MPE Home > Th. List > fvmpti | Structured version Visualization version GIF version |
Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptg.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmpti | ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | fvmptg.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
3 | 1, 2 | fvmptg 6743 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶) |
4 | fvi 6715 | . . . 4 ⊢ (𝐶 ∈ V → ( I ‘𝐶) = 𝐶) | |
5 | 4 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → ( I ‘𝐶) = 𝐶) |
6 | 3, 5 | eqtr4d 2836 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
7 | 1 | eleq1d 2874 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
8 | 2 | dmmpt 6061 | . . . . . . . 8 ⊢ dom 𝐹 = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ V} |
9 | 7, 8 | elrab2 3631 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V)) |
10 | 9 | baib 539 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ dom 𝐹 ↔ 𝐶 ∈ V)) |
11 | 10 | notbid 321 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V)) |
12 | ndmfv 6675 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
13 | 11, 12 | syl6bir 257 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)) |
14 | 13 | imp 410 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
15 | fvprc 6638 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
16 | 15 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅) |
17 | 14, 16 | eqtr4d 2836 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
18 | 6, 17 | pm2.61dan 812 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ↦ cmpt 5110 I cid 5424 dom cdm 5519 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: fvmpt2i 6755 fvmptex 6759 sumeq2ii 15042 summolem3 15063 fsumf1o 15072 isumshft 15186 prodeq2ii 15259 prodmolem3 15279 fprodf1o 15292 |
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