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Mirrors > Home > MPE Home > Th. List > elimasng | Structured version Visualization version GIF version |
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) |
Ref | Expression |
---|---|
elimasng | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4579 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
2 | 1 | imaeq2d 5931 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵})) |
3 | 2 | eleq2d 2900 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵}))) |
4 | opeq1 4805 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝑧〉 = 〈𝐵, 𝑧〉) | |
5 | 4 | eleq1d 2899 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝑦, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝑧〉 ∈ 𝐴)) |
6 | 3, 5 | bibi12d 348 | . 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴))) |
7 | eleq1 2902 | . . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵}))) | |
8 | opeq2 4806 | . . . 4 ⊢ (𝑧 = 𝐶 → 〈𝐵, 𝑧〉 = 〈𝐵, 𝐶〉) | |
9 | 8 | eleq1d 2899 | . . 3 ⊢ (𝑧 = 𝐶 → (〈𝐵, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
10 | 7, 9 | bibi12d 348 | . 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴))) |
11 | vex 3499 | . . 3 ⊢ 𝑦 ∈ V | |
12 | vex 3499 | . . 3 ⊢ 𝑧 ∈ V | |
13 | 11, 12 | elimasn 5956 | . 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) |
14 | 6, 10, 13 | vtocl2g 3574 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 〈cop 4575 “ cima 5560 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: elimasni 5958 eliniseg 5960 inimasn 6015 elpredim 6162 elpredg 6164 dffv3 6668 fvimacnv 6825 fvrnressn 6925 elecg 8334 imasnopn 22300 imasncld 22301 imasncls 22302 ustelimasn 22833 blval2 23174 elbl4 23175 1stpreimas 30443 opelco3 33020 scutval 33267 funpartfv 33408 eltail 33724 elecALTV 35529 brtrclfv2 40079 frege77d 40098 dfafv23 43459 |
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