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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 6216 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) | |
2 | rnnonrel 37897 | . . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 0ss 3972 | . . . . 5 ⊢ ∅ ⊆ {∅} | |
4 | 2, 3 | eqsstri 3635 | . . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} |
5 | ssequn1 3783 | . . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}) | |
6 | 4, 5 | mpbi 220 | . . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅} |
7 | 1, 6 | eleqtri 2699 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} |
8 | fvex 6201 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V | |
9 | 8 | elsn 4192 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅) |
10 | 7, 9 | mpbi 220 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 ◡ccnv 5113 ran crn 5115 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 |
This theorem is referenced by: (None) |
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