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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsetpreimafvb | Structured version Visualization version GIF version |
Description: The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
elsetpreimafvb | ⊢ (𝑆 ∈ 𝑉 → (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 | eleq2i 2831 | . 2 ⊢ (𝑆 ∈ 𝑃 ↔ 𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}) |
3 | eqeq1 2739 | . . . 4 ⊢ (𝑧 = 𝑆 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) | |
4 | 3 | rexbidv 3177 | . . 3 ⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
5 | 4 | elabg 3677 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
6 | 2, 5 | bitrid 283 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 {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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 |
This theorem is referenced by: elsetpreimafv 47310 preimafvelsetpreimafv 47313 0nelsetpreimafv 47315 |
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