Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6765 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶) |
4 | fvi 6739 | . . . 4 ⊢ (𝐶 ∈ V → ( I ‘𝐶) = 𝐶) | |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → ( I ‘𝐶) = 𝐶) |
6 | 3, 5 | eqtr4d 2859 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
7 | 1 | eleq1d 2897 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
8 | 2 | dmmpt 6093 | . . . . . . . 8 ⊢ dom 𝐹 = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ V} |
9 | 7, 8 | elrab2 3682 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V)) |
10 | 9 | baib 538 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ dom 𝐹 ↔ 𝐶 ∈ V)) |
11 | 10 | notbid 320 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V)) |
12 | ndmfv 6699 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
13 | 11, 12 | syl6bir 256 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)) |
14 | 13 | imp 409 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
15 | fvprc 6662 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
16 | 15 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅) |
17 | 14, 16 | eqtr4d 2859 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
18 | 6, 17 | pm2.61dan 811 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ↦ cmpt 5145 I cid 5458 dom cdm 5554 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 |
This theorem is referenced by: fvmpt2i 6777 fvmptex 6781 sumeq2ii 15049 summolem3 15070 fsumf1o 15079 isumshft 15193 prodeq2ii 15266 prodmolem3 15286 fprodf1o 15299 |
Copyright terms: Public domain | W3C validator |