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Mirrors > Home > MPE Home > Th. List > elfv | Structured version Visualization version GIF version |
Description: Membership in a function value. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
elfv | ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fv2 6886 | . . 3 ⊢ (𝐹‘𝐵) = ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)} | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ 𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)}) |
3 | eluniab 4917 | . 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∀wal 1532 ∃wex 1774 ∈ wcel 2099 {cab 2705 ∪ cuni 4903 class class class wbr 5142 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-ss 3962 df-sn 4625 df-uni 4904 df-iota 6494 df-fv 6550 |
This theorem is referenced by: fv3 6909 |
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