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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 3618 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
2 | 1 | imaeq2d 4988 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵})) |
3 | 2 | eleq2d 2259 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵}))) |
4 | opeq1 3793 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝑧〉 = 〈𝐵, 𝑧〉) | |
5 | 4 | eleq1d 2258 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝑦, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝑧〉 ∈ 𝐴)) |
6 | 3, 5 | bibi12d 235 | . 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴))) |
7 | eleq1 2252 | . . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵}))) | |
8 | opeq2 3794 | . . . 4 ⊢ (𝑧 = 𝐶 → 〈𝐵, 𝑧〉 = 〈𝐵, 𝐶〉) | |
9 | 8 | eleq1d 2258 | . . 3 ⊢ (𝑧 = 𝐶 → (〈𝐵, 𝑧〉 ∈ 𝐴 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
10 | 7, 9 | bibi12d 235 | . 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝑧〉 ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴))) |
11 | vex 2755 | . . 3 ⊢ 𝑦 ∈ V | |
12 | vex 2755 | . . 3 ⊢ 𝑧 ∈ V | |
13 | 11, 12 | elimasn 5013 | . 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 〈𝑦, 𝑧〉 ∈ 𝐴) |
14 | 6, 10, 13 | vtocl2g 2816 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {csn 3607 〈cop 3610 “ cima 4647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 |
This theorem is referenced by: eliniseg 5016 inimasn 5064 dffv3g 5530 fvimacnv 5652 funfvima3 5771 elecg 6599 imasnopn 14256 |
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