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Mirrors > Home > ILE Home > Th. List > elrelimasn | GIF version |
Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.) |
Ref | Expression |
---|---|
elrelimasn | ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimag 4975 | . . . . . 6 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵)) | |
2 | 1 | ibi 176 | . . . . 5 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵) |
3 | rexm 3523 | . . . . 5 ⊢ (∃𝑦 ∈ {𝐴}𝑦𝑅𝐵 → ∃𝑦 𝑦 ∈ {𝐴}) | |
4 | elsni 3611 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
5 | 4 | eximi 1600 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝐴} → ∃𝑦 𝑦 = 𝐴) |
6 | 2, 3, 5 | 3syl 17 | . . . 4 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 𝑦 = 𝐴) |
7 | isset 2744 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
8 | 6, 7 | sylibr 134 | . . 3 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V) |
9 | 8 | a1i 9 | . 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V)) |
10 | brrelex1 4666 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
11 | 10 | ex 115 | . 2 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V)) |
12 | imasng 4994 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥}) | |
13 | 12 | eleq2d 2247 | . . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥})) |
14 | brrelex2 4668 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
15 | 14 | ex 115 | . . . . 5 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V)) |
16 | breq2 4008 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
17 | 16 | elab3g 2889 | . . . . 5 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵)) |
18 | 15, 17 | syl 14 | . . . 4 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵)) |
19 | 13, 18 | sylan9bbr 463 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) |
20 | 19 | ex 115 | . 2 ⊢ (Rel 𝑅 → (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))) |
21 | 9, 11, 20 | pm5.21ndd 705 | 1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 Vcvv 2738 {csn 3593 class class class wbr 4004 “ cima 4630 Rel wrel 4632 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-xp 4633 df-rel 4634 df-cnv 4635 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 |
This theorem is referenced by: eliniseg2 5009 |
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