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Mirrors > Home > MPE Home > Th. List > Mathboxes > imasetpreimafvbijlemf | Structured version Visualization version GIF version |
Description: Lemma for imasetpreimafvbij 45688: 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 45678 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ ran 𝐹) |
3 | fnima 6635 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) | |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → (𝐹 “ 𝐴) = ran 𝐹) |
5 | 2, 4 | eleqtrrd 2837 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ (𝐹 “ 𝐴)) |
6 | fundcmpsurinj.h | . 2 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)) | |
7 | 5, 6 | fmptd 7066 | 1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3070 {csn 4590 ∪ cuni 4869 ↦ cmpt 5192 ◡ccnv 5636 ran crn 5638 “ cima 5640 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 |
This theorem is referenced by: imasetpreimafvbijlemf1 45686 imasetpreimafvbijlemfo 45687 |
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