![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvmptnf | Structured version Visualization version GIF version |
Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 6769 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvmptf.1 | ⊢ Ⅎ𝑥𝐴 |
fvmptf.2 | ⊢ Ⅎ𝑥𝐶 |
fvmptf.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptf.4 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptnf | ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptf.4 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
2 | 1 | dmmptss 6062 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 |
3 | 2 | sseli 3911 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) |
4 | eqid 2798 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
5 | 1, 4 | fvmptex 6759 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) |
6 | fvex 6658 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2955 | . . . . . . . . 9 ⊢ Ⅎ𝑥 I | |
9 | fvmptf.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
10 | 8, 9 | nffv 6655 | . . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶) |
11 | fvmptf.3 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)) |
13 | 7, 10, 12, 4 | fvmptf 6766 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
14 | 6, 13 | mpan2 690 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
15 | 5, 14 | syl5eq 2845 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
16 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
17 | 15, 16 | sylan9eq 2853 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
18 | 17 | expcom 417 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅)) |
19 | 3, 18 | syl5 34 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) |
20 | ndmfv 6675 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
21 | 19, 20 | pm2.61d1 183 | 1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 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-csb 3829 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-fn 6327 df-fv 6332 |
This theorem is referenced by: fvmptn 6769 rdgsucmptnf 8048 frsucmptn 8057 |
Copyright terms: Public domain | W3C validator |