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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.) |
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 2979 | . . . . 5 ⊢ Ⅎ𝑦〈[𝑥]𝑅, 𝐴〉 | |
3 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅 | |
4 | nfcsb1v 3909 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfop 4821 | . . . . 5 ⊢ Ⅎ𝑥〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉 |
6 | eceq1 8329 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
7 | csbeq1a 3899 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
8 | 6, 7 | opeq12d 4813 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈[𝑥]𝑅, 𝐴〉 = 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
9 | 2, 5, 8 | cbvmpt 5169 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
10 | 9 | rneqi 5809 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
11 | 1, 10 | eqtri 2846 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
12 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
13 | 12 | ralrimiva 3184 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
14 | 4 | nfel1 2996 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌 |
15 | 7 | eleq1d 2899 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
16 | 14, 15 | rspc 3613 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
17 | 13, 16 | mpan9 509 | . 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 3888 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
21 | 11, 17, 18, 19, 20 | qliftfun 8384 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⦋csb 3885 〈cop 4575 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 Fun wfun 6351 Er wer 8288 [cec 8289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-er 8291 df-ec 8293 df-qs 8297 |
This theorem is referenced by: (None) |
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