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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimafvsspwdm | Structured version Visualization version GIF version |
Description: The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
preimafvsspwdm | ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setpreimafvex.p | . . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
2 | 1 | elsetpreimafvssdm 43595 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → 𝑠 ⊆ 𝐴) |
3 | 2 | ralrimiva 3182 | . 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴) |
4 | pwssb 5023 | . 2 ⊢ (𝑃 ⊆ 𝒫 𝐴 ↔ ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {cab 2799 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ◡ccnv 5554 “ cima 5558 Fn wfn 6350 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fn 6358 |
This theorem is referenced by: (None) |
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