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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 6201.) 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 3536 | . . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) | |
2 | fvmptex.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3549 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3542 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 4749 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2644 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmpti 6281 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
9 | 1 | fveq2d 6195 | . . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
10 | fvmptex.2 | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) | |
11 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵) | |
12 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑥 I | |
13 | 12, 4 | nffv 6198 | . . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵) |
14 | 5 | fveq2d 6195 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
15 | 11, 13, 14 | cbvmpt 4749 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
16 | 10, 15 | eqtri 2644 | . . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵)) |
17 | fvex 6201 | . . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V | |
18 | 9, 16, 17 | fvmpt 6282 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵)) |
19 | 8, 18 | eqtr4d 2659 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶)) |
20 | 2 | dmmptss 5631 | . . . . . 6 ⊢ dom 𝐹 ⊆ 𝐴 |
21 | 20 | sseli 3599 | . . . . 5 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴) |
22 | 21 | con3i 150 | . . . 4 ⊢ (¬ 𝐶 ∈ 𝐴 → ¬ 𝐶 ∈ dom 𝐹) |
23 | ndmfv 6218 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅) |
25 | fvex 6201 | . . . . . 6 ⊢ ( I ‘𝐵) ∈ V | |
26 | 25, 10 | dmmpti 6023 | . . . . 5 ⊢ dom 𝐺 = 𝐴 |
27 | 26 | eleq2i 2693 | . . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴) |
28 | ndmfv 6218 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅) | |
29 | 27, 28 | sylnbir 321 | . . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅) |
30 | 24, 29 | eqtr4d 2659 | . 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 1483 ∈ wcel 1990 ⦋csb 3533 ∅c0 3915 ↦ cmpt 4729 I cid 5023 dom cdm 5114 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: fvmptnf 6302 sumeq2ii 14423 prodeq2ii 14643 |
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