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Mirrors > Home > MPE Home > Th. List > elima3 | Structured version Visualization version GIF version |
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
elima.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elima3 | ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elima.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | elima2 6055 | . 2 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴)) |
3 | df-br 5139 | . . . 4 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵) | |
4 | 3 | anbi2i 622 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵)) |
5 | 4 | exbii 1842 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵)) |
6 | 2, 5 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 ⟨cop 4626 class class class wbr 5138 “ cima 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-cnv 5674 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 |
This theorem is referenced by: cnvresima 6219 imaiun 7236 1stpreimas 32396 elima4 35242 imaiun1 42891 snhesn 43026 |
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