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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvnonrel | Structured version Visualization version GIF version |
Description: The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.) |
Ref | Expression |
---|---|
fvnonrel | ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvrn0 6915 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) | |
2 | rnnonrel 42918 | . . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 0ss 4391 | . . . . 5 ⊢ ∅ ⊆ {∅} | |
4 | 2, 3 | eqsstri 4011 | . . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} |
5 | ssequn1 4175 | . . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅} |
7 | 1, 6 | eleqtri 2825 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} |
8 | fvex 6898 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V | |
9 | 8 | elsn 4638 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅) |
10 | 7, 9 | mpbi 229 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∪ cun 3941 ⊆ wss 3943 ∅c0 4317 {csn 4623 ◡ccnv 5668 ran crn 5670 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-iota 6489 df-fv 6545 |
This theorem is referenced by: (None) |
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