![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fniniseg2 | Structured version Visualization version GIF version |
Description: Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
fniniseg2 | ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fncnvima2 7062 | . 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}}) | |
2 | fvex 6904 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
3 | 2 | elsn 4643 | . . 3 ⊢ ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵) |
4 | 3 | rabbii 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵} |
5 | 1, 4 | eqtrdi 2787 | 1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3431 {csn 4628 ◡ccnv 5675 “ cima 5679 Fn wfn 6538 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: qusker 32900 ply1annidllem 33217 algextdeglem6 33233 idomrootle 42400 proot1hash 42405 |
Copyright terms: Public domain | W3C validator |