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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 47376 | . 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 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {csn 4625 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 | 
| This theorem is referenced by: elsetpreimafvssdm 47378 fvelsetpreimafv 47379 elsetpreimafvbi 47383 imasetpreimafvbijlemfo 47397 | 
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