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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 6086 | . 2 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴)) |
3 | df-br 5149 | . . . 4 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | anbi2i 623 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ (𝑥 ∈ 𝐶 ∧ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
5 | 4 | exbii 1845 | . 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 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 〈cop 4637 class class class wbr 5148 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: cnvresima 6252 imaiun 7265 1stpreimas 32721 elima4 35757 imaiun1 43641 snhesn 43776 |
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