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Mirrors > Home > MPE Home > Th. List > qliftfuns | Structured version Visualization version GIF version |
Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.) |
Ref | Expression |
---|---|
qlift.1 | ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) |
qlift.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) |
qlift.3 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
qlift.4 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
qliftfuns | ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qlift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) | |
2 | nfcv 2908 | . . . . 5 ⊢ Ⅎ𝑦⟨[𝑥]𝑅, 𝐴⟩ | |
3 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅 | |
4 | nfcsb1v 3881 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfop 4847 | . . . . 5 ⊢ Ⅎ𝑥⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩ |
6 | eceq1 8687 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
7 | csbeq1a 3870 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
8 | 6, 7 | opeq12d 4839 | . . . . 5 ⊢ (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩) |
9 | 2, 5, 8 | cbvmpt 5217 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩) |
10 | 9 | rneqi 5893 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩) |
11 | 1, 10 | eqtri 2765 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩) |
12 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
13 | 12 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
14 | 4 | nfel1 2924 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌 |
15 | 7 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
16 | 14, 15 | rspc 3570 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
17 | 13, 16 | mpan9 508 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌) |
18 | qlift.3 | . 2 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
19 | qlift.4 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
20 | csbeq1 3859 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
21 | 11, 17, 18, 19, 20 | qliftfun 8742 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ⦋csb 3856 ⟨cop 4593 class class class wbr 5106 ↦ cmpt 5189 ran crn 5635 Fun wfun 6491 Er wer 8646 [cec 8647 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-er 8649 df-ec 8651 df-qs 8655 |
This theorem is referenced by: (None) |
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