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 6792 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 6095 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 |
3 | 2 | sseli 3963 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) |
4 | eqid 2821 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
5 | 1, 4 | fvmptex 6782 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) |
6 | fvex 6683 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑥 I | |
9 | fvmptf.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
10 | 8, 9 | nffv 6680 | . . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶) |
11 | fvmptf.3 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | fveq2d 6674 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)) |
13 | 7, 10, 12, 4 | fvmptf 6789 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
14 | 6, 13 | mpan2 689 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
15 | 5, 14 | syl5eq 2868 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
16 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
17 | 15, 16 | sylan9eq 2876 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
18 | 17 | expcom 416 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅)) |
19 | 3, 18 | syl5 34 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) |
20 | ndmfv 6700 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
21 | 19, 20 | pm2.61d1 182 | 1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 Vcvv 3494 ∅c0 4291 ↦ cmpt 5146 I cid 5459 dom cdm 5555 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: fvmptn 6792 rdgsucmptnf 8065 frsucmptn 8074 |
Copyright terms: Public domain | W3C validator |