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Mirrors > Home > MPE Home > Th. List > elrelimasn | Structured version Visualization version 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 | relimasn 6083 | . . 3 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥}) | |
2 | 1 | eleq2d 2819 | . 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥})) |
3 | brrelex2 5730 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
4 | 3 | ex 413 | . . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V)) |
5 | breq2 5152 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
6 | 5 | elab3g 3675 | . . 3 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵)) |
8 | 2, 7 | bitrd 278 | 1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 {cab 2709 Vcvv 3474 {csn 4628 class class class wbr 5148 “ cima 5679 Rel wrel 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: eliniseg2 6105 dprd2dlem2 19909 dprd2dlem1 19910 dprd2da 19911 dprd2d2 19913 dpjfval 19924 ustuqtop4 23748 utop2nei 23754 utop3cls 23755 ucncn 23789 extdgval 32728 cnambfre 36531 frege133d 42506 nzin 43067 |
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