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| Mirrors > Home > MPE Home > Th. List > mptiniseg | Structured version Visualization version GIF version | ||
| Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| dmmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| Ref | Expression | 
|---|---|
| mptiniseg | ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | 1 | mptpreima 6258 | . 2 ⊢ (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}} | 
| 3 | elsn2g 4664 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶)) | |
| 4 | 3 | rabbidv 3444 | . 2 ⊢ (𝐶 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}} = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶}) | 
| 5 | 2, 4 | eqtrid 2789 | 1 ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 {csn 4626 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 | 
| This theorem is referenced by: ramub1lem1 17064 frlmsslss 21794 symgtgp 24114 csscld 25283 clsocv 25284 sqff1o 27225 dchrfi 27299 poimirlem30 37657 ftc1anclem6 37705 pwssplit4 43101 pwslnmlem2 43105 | 
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