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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 6126 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) | |
2 | rnnonrel 36916 | . . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 0ss 3924 | . . . . 5 ⊢ ∅ ⊆ {∅} | |
4 | 2, 3 | eqsstri 3598 | . . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} |
5 | ssequn1 3745 | . . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}) | |
6 | 4, 5 | mpbi 219 | . . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅} |
7 | 1, 6 | eleqtri 2686 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} |
8 | fvex 6113 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V | |
9 | 8 | elsn 4140 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅) |
10 | 7, 9 | mpbi 219 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 ◡ccnv 5037 ran crn 5039 ‘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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 |
This theorem is referenced by: (None) |
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