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Mirrors > Home > MPE Home > Th. List > acnlem | Structured version Visualization version GIF version |
Description: Construct a mapping satisfying the consequent of isacn 10039. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acnlem | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvssunirn 6925 | . . . . . 6 ⊢ (𝑓‘𝑥) ⊆ ∪ ran 𝑓 | |
2 | simpr 486 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ (𝑓‘𝑥)) | |
3 | 1, 2 | sselid 3981 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ ∪ ran 𝑓) |
4 | 3 | ralimiaa 3083 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓) |
5 | eqid 2733 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fmpt 7110 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓) |
7 | 4, 6 | sylib 217 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓) |
8 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
9 | vex 3479 | . . . . . 6 ⊢ 𝑓 ∈ V | |
10 | 9 | rnex 7903 | . . . . 5 ⊢ ran 𝑓 ∈ V |
11 | 10 | uniex 7731 | . . . 4 ⊢ ∪ ran 𝑓 ∈ V |
12 | fex2 7924 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
13 | 11, 12 | mp3an3 1451 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
14 | 7, 8, 13 | syl2anr 598 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
15 | 5 | fvmpt2 7010 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
16 | 15, 2 | eqeltrd 2834 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)) |
17 | 16 | ralimiaa 3083 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)) |
18 | 17 | adantl 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)) |
19 | nfmpt1 5257 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
20 | 19 | nfeq2 2921 | . . 3 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
21 | fveq1 6891 | . . . 4 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) | |
22 | 21 | eleq1d 2819 | . . 3 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))) |
23 | 20, 22 | ralbid 3271 | . 2 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))) |
24 | 14, 18, 23 | spcedv 3589 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∪ cuni 4909 ↦ cmpt 5232 ran crn 5678 ⟶wf 6540 ‘cfv 6544 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 |
This theorem is referenced by: numacn 10044 acndom 10046 acndom2 10049 |
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