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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 6211 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 5548 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 |
3 | 2 | sseli 3564 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) |
4 | eqid 2610 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
5 | 1, 4 | fvmptex 6203 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) |
6 | fvex 6113 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2751 | . . . . . . . . 9 ⊢ Ⅎ𝑥 I | |
9 | fvmptf.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
10 | 8, 9 | nffv 6110 | . . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶) |
11 | fvmptf.3 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)) |
13 | 7, 10, 12, 4 | fvmptf 6209 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
14 | 6, 13 | mpan2 703 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
15 | 5, 14 | syl5eq 2656 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
16 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
17 | 15, 16 | sylan9eq 2664 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
18 | 17 | expcom 450 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅)) |
19 | 3, 18 | syl5 33 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) |
20 | ndmfv 6128 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
21 | 19, 20 | pm2.61d1 170 | 1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 I cid 4948 dom cdm 5038 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: fvmptn 6211 rdgsucmptnf 7412 frsucmptn 7421 |
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