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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 6556 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 5876 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 |
3 | 2 | sseli 3823 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) |
4 | eqid 2825 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
5 | 1, 4 | fvmptex 6546 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) |
6 | fvex 6450 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2969 | . . . . . . . . 9 ⊢ Ⅎ𝑥 I | |
9 | fvmptf.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
10 | 8, 9 | nffv 6447 | . . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶) |
11 | fvmptf.3 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | fveq2d 6441 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)) |
13 | 7, 10, 12, 4 | fvmptf 6553 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
14 | 6, 13 | mpan2 682 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶)) |
15 | 5, 14 | syl5eq 2873 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
16 | fvprc 6430 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
17 | 15, 16 | sylan9eq 2881 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
18 | 17 | expcom 404 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅)) |
19 | 3, 18 | syl5 34 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)) |
20 | ndmfv 6467 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
21 | 19, 20 | pm2.61d1 173 | 1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1656 ∈ wcel 2164 Ⅎwnfc 2956 Vcvv 3414 ∅c0 4146 ↦ cmpt 4954 I cid 5251 dom cdm 5346 ‘cfv 6127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-fv 6135 |
This theorem is referenced by: fvmptn 6556 rdgsucmptnf 7796 frsucmptn 7805 |
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