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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 7040 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 6262 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐷 |
3 | 2 | sseli 3990 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷) |
4 | eqid 2734 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) | |
5 | 1, 4 | fvmptex 7029 | . . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) |
6 | fvex 6919 | . . . . . . 7 ⊢ ( I ‘𝐶) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2902 | . . . . . . . . 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 7036 | . . . . . . 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 2786 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
16 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
17 | 15, 16 | sylan9eq 2794 | . . . 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 1536 ∈ wcel 2105 Ⅎwnfc 2887 Vcvv 3477 ∅c0 4338 ↦ cmpt 5230 I cid 5581 dom cdm 5688 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: fvmptn 7040 rdgsucmptnf 8467 frsucmptn 8477 |
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