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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 3543 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
2 | 1 | imaeq2d 4889 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵})) |
3 | 2 | eleq2d 2210 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵}))) |
4 | opeq1 3713 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝑧〉 = 〈𝐵, 𝑧〉) | |
5 | 4 | eleq1d 2209 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝑦, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝑧〉 ∈ 𝐴)) |
6 | 3, 5 | bibi12d 234 | . 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴))) |
7 | eleq1 2203 | . . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵}))) | |
8 | opeq2 3714 | . . . 4 ⊢ (𝑧 = 𝐶 → 〈𝐵, 𝑧〉 = 〈𝐵, 𝐶〉) | |
9 | 8 | eleq1d 2209 | . . 3 ⊢ (𝑧 = 𝐶 → (〈𝐵, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
10 | 7, 9 | bibi12d 234 | . 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴))) |
11 | vex 2692 | . . 3 ⊢ 𝑦 ∈ V | |
12 | vex 2692 | . . 3 ⊢ 𝑧 ∈ V | |
13 | 11, 12 | elimasn 4914 | . 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) |
14 | 6, 10, 13 | vtocl2g 2753 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {csn 3532 〈cop 3535 “ cima 4550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: eliniseg 4917 inimasn 4964 dffv3g 5425 fvimacnv 5543 funfvima3 5659 elecg 6475 imasnopn 12507 |
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