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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 2903 | . 2 ⊢ (𝑆 ∈ 𝑃 ↔ 𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}) |
3 | eqeq1 2824 | . . . 4 ⊢ (𝑧 = 𝑆 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) | |
4 | 3 | rexbidv 3296 | . . 3 ⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
5 | 4 | elabg 3662 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
6 | 2, 5 | syl5bb 285 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 {cab 2798 ∃wrex 3138 {csn 4560 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 |
This theorem is referenced by: elsetpreimafv 43619 preimafvelsetpreimafv 43622 0nelsetpreimafv 43624 |
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