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| Mirrors > Home > MPE Home > Th. List > Mathboxes > imasetpreimafvbijlemf | Structured version Visualization version GIF version | ||
| Description: Lemma for imasetpreimafvbij 47393: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.) |
| Ref | Expression |
|---|---|
| fundcmpsurinj.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
| fundcmpsurinj.h | ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) |
| Ref | Expression |
|---|---|
| imasetpreimafvbijlemf | ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fundcmpsurinj.p | . . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
| 2 | 1 | uniimaelsetpreimafv 47383 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ ran 𝐹) |
| 3 | fnima 6698 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) | |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → (𝐹 “ 𝐴) = ran 𝐹) |
| 5 | 2, 4 | eleqtrrd 2844 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ (𝐹 “ 𝐴)) |
| 6 | fundcmpsurinj.h | . 2 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) | |
| 7 | 5, 6 | fmptd 7134 | 1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: imasetpreimafvbijlemf1 47391 imasetpreimafvbijlemfo 47392 |
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