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Mirrors > Home > ILE Home > Th. List > qliftfuns | 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 2219 | . . . . 5 ⊢ Ⅎ𝑦〈[𝑥]𝑅, 𝐴〉 | |
3 | nfcv 2219 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅 | |
4 | nfcsb1v 2938 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfop 3586 | . . . . 5 ⊢ Ⅎ𝑥〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉 |
6 | eceq1 6164 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
7 | csbeq1a 2916 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
8 | 6, 7 | opeq12d 3578 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈[𝑥]𝑅, 𝐴〉 = 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
9 | 2, 5, 8 | cbvmpt 3872 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
10 | 9 | rneqi 4580 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
11 | 1, 10 | eqtri 2101 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
12 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
13 | 12 | ralrimiva 2434 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
14 | 4 | nfel1 2229 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌 |
15 | 7 | eleq1d 2147 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
16 | 14, 15 | rspc 2695 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
17 | 13, 16 | mpan9 275 | . 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 2911 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
21 | 11, 17, 18, 19, 20 | qliftfun 6211 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 ∀wral 2348 Vcvv 2601 ⦋csb 2908 〈cop 3401 class class class wbr 3785 ↦ cmpt 3839 ran crn 4364 Fun wfun 4916 Er wer 6126 [cec 6127 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-er 6129 df-ec 6131 df-qs 6135 |
This theorem is referenced by: (None) |
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