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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > setpreimafvex | Structured version Visualization version GIF version |
Description: The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
setpreimafvex | ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setpreimafvex.p | . 2 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
2 | abrexexg 7958 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V) | |
3 | 1, 2 | eqeltrid 2832 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2704 ∃wrex 3065 Vcvv 3469 {csn 4624 ◡ccnv 5671 “ cima 5675 ‘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-ext 2698 ax-rep 5279 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-rex 3066 df-v 3471 |
This theorem is referenced by: fundcmpsurbijinjpreimafv 46670 |
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