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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 5391 | . 2 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴)) |
3 | df-br 4584 | . . . 4 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | anbi2i 726 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ (𝑥 ∈ 𝐶 ∧ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
5 | 4 | exbii 1764 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
6 | 2, 5 | bitri 263 | 1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 〈𝑥, 𝐴〉 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: cnvresima 5541 imaiun 6407 1stpreimas 28866 elima4 30924 imaiun1 36962 snhesn 37100 |
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