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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsetpreimafv | Structured version Visualization version GIF version |
Description: An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
elsetpreimafv | ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setpreimafvex.p | . . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
2 | 1 | elsetpreimafvb 46724 | . 2 ⊢ (𝑆 ∈ 𝑃 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
3 | 2 | ibi 267 | 1 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2705 ∃wrex 3067 {csn 4629 ◡ccnv 5677 “ cima 5681 ‘cfv 6548 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rex 3068 |
This theorem is referenced by: elsetpreimafvssdm 46726 fvelsetpreimafv 46727 elsetpreimafvbi 46731 imasetpreimafvbijlemfo 46745 |
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