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| 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 7041 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 6261 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 | 
| 3 | 2 | sseli 3979 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) | 
| 4 | eqid 2737 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
| 5 | 1, 4 | fvmptex 7030 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) | 
| 6 | fvex 6919 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
| 7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
| 8 | nfcv 2905 | . . . . . . . . 9 ⊢ Ⅎ𝑥 I | |
| 9 | fvmptf.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
| 10 | 8, 9 | nffv 6916 | . . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶) | 
| 11 | fvmptf.3 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 12 | 11 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)) | 
| 13 | 7, 10, 12, 4 | fvmptf 7037 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) | 
| 14 | 6, 13 | mpan2 691 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) | 
| 15 | 5, 14 | eqtrid 2789 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) | 
| 16 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
| 17 | 15, 16 | sylan9eq 2797 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) | 
| 18 | 17 | expcom 413 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅)) | 
| 19 | 3, 18 | syl5 34 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) | 
| 20 | ndmfv 6941 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
| 21 | 19, 20 | pm2.61d1 180 | 1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 I cid 5577 dom cdm 5685 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 | 
| This theorem is referenced by: fvmptn 7041 rdgsucmptnf 8469 frsucmptn 8479 | 
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