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Mirrors > Home > MPE Home > Th. List > Mathboxes > imasetpreimafvbijlemfv1 | Structured version Visualization version GIF version |
Description: Lemma for imasetpreimafvbij 46669: 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 46653 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃) |
3 | elnelne2 3053 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅) | |
4 | 3 | expcom 413 | . . . 4 ⊢ (∅ ∉ 𝑃 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅)) |
6 | 5 | imp 406 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ ∅) |
7 | simpr 484 | . . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) | |
8 | fundcmpsurinj.h | . . . . . . . 8 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) | |
9 | 1, 8 | imasetpreimafvbijlemfv 46665 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃 ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦)) |
10 | 9 | 3expa 1116 | . . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦)) |
11 | 7, 10 | jca 511 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))) |
12 | 11 | ex 412 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))) |
13 | 12 | eximdv 1913 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (∃𝑦 𝑦 ∈ 𝑋 → ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))) |
14 | n0 4342 | . . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋) | |
15 | df-rex 3066 | . . 3 ⊢ (∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))) | |
16 | 13, 14, 15 | 3imtr4g 296 | . 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 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2704 ≠ wne 2935 ∉ wnel 3041 ∃wrex 3065 ∅c0 4318 {csn 4624 ∪ cuni 4903 ↦ cmpt 5225 ◡ccnv 5671 “ cima 5675 Fn wfn 6537 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 |
This theorem is referenced by: imasetpreimafvbijlemf1 46667 |
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