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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 5044 and to prove elimasn1 5944 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 488 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ 𝑊) | |
2 | imasng 5940 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) |
4 | simpr 488 | . . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶) | |
5 | 4 | breq2d 5055 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶)) |
6 | 1, 3, 5 | elabd2 3572 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {cab 2712 {csn 4531 class class class wbr 5043 “ cima 5543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 |
This theorem is referenced by: elimasn1 5944 elimasng 5945 elimasni 5948 elinisegg 5950 |
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