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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 6937 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) | |
2 | rnnonrel 43581 | . . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 0ss 4406 | . . . . 5 ⊢ ∅ ⊆ {∅} | |
4 | 2, 3 | eqsstri 4030 | . . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} |
5 | ssequn1 4196 | . . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}) | |
6 | 4, 5 | mpbi 230 | . . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅} |
7 | 1, 6 | eleqtri 2837 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} |
8 | fvex 6920 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V | |
9 | 8 | elsn 4646 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅) |
10 | 7, 9 | mpbi 230 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {csn 4631 ◡ccnv 5688 ran crn 5690 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-iota 6516 df-fv 6571 |
This theorem is referenced by: (None) |
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