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| Mirrors > Home > MPE Home > Th. List > elimasng1 | Structured version Visualization version GIF version | ||
| Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5144 and to prove elimasn1 6106 from it. (Revised by BJ, 16-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| elimasng1 | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpr 484 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ 𝑊) | |
| 2 | imasng 6102 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | |
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | 
| 4 | simpr 484 | . . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶) | |
| 5 | 4 | breq2d 5155 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶)) | 
| 6 | 1, 3, 5 | elabd2 3670 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {csn 4626 class class class wbr 5143 “ 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 | 
| This theorem is referenced by: elimasn1 6106 elimasng 6107 elimasni 6109 elinisegg 6111 | 
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