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Mirrors > Home > MPE Home > Th. List > fvmptex | Structured version Visualization version GIF version |
Description: Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6342.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fvmptex.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
fvmptex.2 | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) |
Ref | Expression |
---|---|
fvmptex | ⊢ (𝐹‘𝐶) = (𝐺‘𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3683 | . . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) | |
2 | fvmptex.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | nfcv 2912 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3696 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3689 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 4881 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2792 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmpti 6423 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
9 | 1 | fveq2d 6336 | . . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
10 | fvmptex.2 | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) | |
11 | nfcv 2912 | . . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵) | |
12 | nfcv 2912 | . . . . . . 7 ⊢ Ⅎ𝑥 I | |
13 | 12, 4 | nffv 6339 | . . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵) |
14 | 5 | fveq2d 6336 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
15 | 11, 13, 14 | cbvmpt 4881 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
16 | 10, 15 | eqtri 2792 | . . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
17 | fvex 6342 | . . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V | |
18 | 9, 16, 17 | fvmpt 6424 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
19 | 8, 18 | eqtr4d 2807 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶)) |
20 | 2 | dmmptss 5775 | . . . . . 6 ⊢ dom 𝐹 ⊆ 𝐴 |
21 | 20 | sseli 3746 | . . . . 5 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴) |
22 | 21 | con3i 151 | . . . 4 ⊢ (¬ 𝐶 ∈ 𝐴 → ¬ 𝐶 ∈ dom 𝐹) |
23 | ndmfv 6359 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅) |
25 | fvex 6342 | . . . . . 6 ⊢ ( I ‘𝐵) ∈ V | |
26 | 25, 10 | dmmpti 6163 | . . . . 5 ⊢ dom 𝐺 = 𝐴 |
27 | 26 | eleq2i 2841 | . . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴) |
28 | ndmfv 6359 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅) | |
29 | 27, 28 | sylnbir 320 | . . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅) |
30 | 24, 29 | eqtr4d 2807 | . 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶)) |
31 | 19, 30 | pm2.61i 176 | 1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1630 ∈ wcel 2144 ⦋csb 3680 ∅c0 4061 ↦ cmpt 4861 I cid 5156 dom cdm 5249 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-fv 6039 |
This theorem is referenced by: fvmptnf 6444 sumeq2ii 14630 prodeq2ii 14849 |
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