![]() |
Mathbox for BTernaryTau |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > gblacfnacd | Structured version Visualization version GIF version |
Description: If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10160) holds. Note that 𝐹 must be a proper class by fndmexb 7929. This means we cannot show that the existence of a class that behaves as a global choice function is sufficient because we only have existential quantifiers for sets, not (proper) classes. However, if a class variant of exlimiv 1928 were available, then it could be used alongside the closed form of this theorem to prove that result. (Contributed by BTernaryTau, 12-Dec-2024.) |
Ref | Expression |
---|---|
gblacfnacd.1 | ⊢ (𝜑 → 𝐹 Fn V) |
gblacfnacd.2 | ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) |
Ref | Expression |
---|---|
gblacfnacd | ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gblacfnacd.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn V) | |
2 | fnfun 6669 | . . . 4 ⊢ (𝐹 Fn V → Fun 𝐹) | |
3 | resfunexg 7235 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V) | |
4 | 3 | elvd 3484 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝑥) ∈ V) |
5 | 1, 2, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑥) ∈ V) |
6 | ssv 4020 | . . . . 5 ⊢ 𝑥 ⊆ V | |
7 | fnssres 6692 | . . . . 5 ⊢ ((𝐹 Fn V ∧ 𝑥 ⊆ V) → (𝐹 ↾ 𝑥) Fn 𝑥) | |
8 | 1, 6, 7 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑥) Fn 𝑥) |
9 | gblacfnacd.2 | . . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) | |
10 | 9 | 19.21bi 2187 | . . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) |
11 | fvres 6926 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑧) = (𝐹‘𝑧)) | |
12 | 11 | eleq1d 2824 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧 ↔ (𝐹‘𝑧) ∈ 𝑧)) |
13 | 12 | imbi2d 340 | . . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))) |
14 | 10, 13 | syl5ibrcom 247 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))) |
15 | 14 | ralrimiv 3143 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)) |
16 | 8, 15 | jca 511 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))) |
17 | fneq1 6660 | . . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐹 ↾ 𝑥) Fn 𝑥)) | |
18 | fveq1 6906 | . . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐹 ↾ 𝑥)‘𝑧)) | |
19 | 18 | eleq1d 2824 | . . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)) |
20 | 19 | imbi2d 340 | . . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))) |
21 | 20 | ralbidv 3176 | . . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))) |
22 | 17, 21 | anbi12d 632 | . . 3 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))) |
23 | 5, 16, 22 | spcedv 3598 | . 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |
24 | 23 | alrimiv 1925 | 1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |