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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvelsetpreimafv | Structured version Visualization version GIF version |
Description: There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
fvelsetpreimafv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimafvsnel 45645 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})) | |
2 | 1 | adantrr 716 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})) |
3 | eleq2 2827 | . . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))) | |
4 | 3 | ad2antll 728 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))) |
5 | 2, 4 | mpbird 257 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ 𝑆) |
6 | simprr 772 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) | |
7 | 5, 6 | jca 513 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
8 | 7 | ex 414 | . . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))) |
9 | 8 | reximdv2 3162 | . 2 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) |
10 | setpreimafvex.p | . . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
11 | 10 | elsetpreimafv 45651 | . 2 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
12 | 9, 11 | impel 507 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∃wrex 3074 {csn 4591 ◡ccnv 5637 “ cima 5641 Fn wfn 6496 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 |
This theorem is referenced by: imaelsetpreimafv 45661 |
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