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Mirrors > Home > MPE Home > Th. List > elsnres | Structured version Visualization version GIF version |
Description: Membership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.) |
Ref | Expression |
---|---|
elsnres.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elsnres | ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elres 5435 | . 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | rexcom4 3225 | . 2 ⊢ (∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
3 | elsnres.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | opeq1 4402 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝑦〉 = 〈𝐶, 𝑦〉) | |
5 | 4 | eqeq2d 2632 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝐶, 𝑦〉)) |
6 | 4 | eleq1d 2686 | . . . . 5 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
7 | 5, 6 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵))) |
8 | 3, 7 | rexsn 4223 | . . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1774 | . 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
10 | 1, 2, 9 | 3bitri 286 | 1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 {csn 4177 〈cop 4183 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-res 5126 |
This theorem is referenced by: fvn0ssdmfun 6350 frxp 7287 |
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