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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 4326 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
2 | 1 | imaeq2d 5607 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵})) |
3 | 2 | eleq2d 2836 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵}))) |
4 | opeq1 4539 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝑧〉 = 〈𝐵, 𝑧〉) | |
5 | 4 | eleq1d 2835 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝑦, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝑧〉 ∈ 𝐴)) |
6 | 3, 5 | bibi12d 334 | . 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴))) |
7 | eleq1 2838 | . . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵}))) | |
8 | opeq2 4540 | . . . 4 ⊢ (𝑧 = 𝐶 → 〈𝐵, 𝑧〉 = 〈𝐵, 𝐶〉) | |
9 | 8 | eleq1d 2835 | . . 3 ⊢ (𝑧 = 𝐶 → (〈𝐵, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
10 | 7, 9 | bibi12d 334 | . 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴))) |
11 | vex 3354 | . . 3 ⊢ 𝑦 ∈ V | |
12 | vex 3354 | . . 3 ⊢ 𝑧 ∈ V | |
13 | 11, 12 | elimasn 5631 | . 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) |
14 | 6, 10, 13 | vtocl2g 3421 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {csn 4316 〈cop 4322 “ cima 5252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 |
This theorem is referenced by: elimasni 5633 eliniseg 5635 inimasn 5691 elpredim 5835 elpredg 5837 dffv3 6328 fvimacnv 6475 fvrnressn 6571 elecg 7937 imasnopn 21714 imasncld 21715 imasncls 21716 ustelimasn 22246 blval2 22587 elbl4 22588 1stpreimas 29823 opelco3 32014 scutval 32248 funpartfv 32389 eltail 32706 elecALTV 34373 brtrclfv2 38545 frege77d 38564 |
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