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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 7984 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V) | |
3 | 1, 2 | eqeltrid 2843 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 Vcvv 3478 {csn 4631 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 |
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-rep 5285 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-v 3480 |
This theorem is referenced by: fundcmpsurbijinjpreimafv 47332 |
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