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Mirrors > Home > MPE Home > Th. List > Mathboxes > imasetpreimafvbijlemfv1 | Structured version Visualization version GIF version |
Description: Lemma for imasetpreimafvbij 43641: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.) |
Ref | Expression |
---|---|
fundcmpsurinj.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
fundcmpsurinj.h | ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) |
Ref | Expression |
---|---|
imasetpreimafvbijlemfv1 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fundcmpsurinj.p | . . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
2 | 1 | 0nelsetpreimafv 43625 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃) |
3 | elnelne2 3133 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅) | |
4 | 3 | expcom 416 | . . . 4 ⊢ (∅ ∉ 𝑃 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅)) |
6 | 5 | imp 409 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ ∅) |
7 | simpr 487 | . . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) | |
8 | fundcmpsurinj.h | . . . . . . . 8 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) | |
9 | 1, 8 | imasetpreimafvbijlemfv 43637 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃 ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦)) |
10 | 9 | 3expa 1113 | . . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦)) |
11 | 7, 10 | jca 514 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))) |
12 | 11 | ex 415 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))) |
13 | 12 | eximdv 1917 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (∃𝑦 𝑦 ∈ 𝑋 → ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))) |
14 | n0 4303 | . . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋) | |
15 | df-rex 3143 | . . 3 ⊢ (∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))) | |
16 | 13, 14, 15 | 3imtr4g 298 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑋 ≠ ∅ → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))) |
17 | 6, 16 | mpd 15 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 {cab 2798 ≠ wne 3015 ∉ wnel 3122 ∃wrex 3138 ∅c0 4284 {csn 4560 ∪ cuni 4831 ↦ cmpt 5139 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 ‘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 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: imasetpreimafvbijlemf1 43639 |
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