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Mirrors > Home > ILE Home > Th. List > elimasng | 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 3386 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
2 | 1 | imaeq2d 4668 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵})) |
3 | 2 | eleq2d 2107 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵}))) |
4 | opeq1 3549 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝑧〉 = 〈𝐵, 𝑧〉) | |
5 | 4 | eleq1d 2106 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝑦, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝑧〉 ∈ 𝐴)) |
6 | 3, 5 | bibi12d 224 | . 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴))) |
7 | eleq1 2100 | . . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵}))) | |
8 | opeq2 3550 | . . . 4 ⊢ (𝑧 = 𝐶 → 〈𝐵, 𝑧〉 = 〈𝐵, 𝐶〉) | |
9 | 8 | eleq1d 2106 | . . 3 ⊢ (𝑧 = 𝐶 → (〈𝐵, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
10 | 7, 9 | bibi12d 224 | . 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴))) |
11 | vex 2560 | . . 3 ⊢ 𝑦 ∈ V | |
12 | vex 2560 | . . 3 ⊢ 𝑧 ∈ V | |
13 | 11, 12 | elimasn 4692 | . 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) |
14 | 6, 10, 13 | vtocl2g 2617 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 {csn 3375 〈cop 3378 “ cima 4348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: eliniseg 4695 inimasn 4741 dffv3g 5174 fvimacnv 5282 funfvima3 5392 elecg 6144 |
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