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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 2282 | . . . . 5 ⊢ Ⅎ𝑦〈[𝑥]𝑅, 𝐴〉 | |
3 | nfcv 2282 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅 | |
4 | nfcsb1v 3040 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfop 3729 | . . . . 5 ⊢ Ⅎ𝑥〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉 |
6 | eceq1 6472 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
7 | csbeq1a 3016 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
8 | 6, 7 | opeq12d 3721 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈[𝑥]𝑅, 𝐴〉 = 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
9 | 2, 5, 8 | cbvmpt 4031 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
10 | 9 | rneqi 4775 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
11 | 1, 10 | eqtri 2161 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
12 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
13 | 12 | ralrimiva 2508 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
14 | 4 | nfel1 2293 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌 |
15 | 7 | eleq1d 2209 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
16 | 14, 15 | rspc 2787 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
17 | 13, 16 | mpan9 279 | . 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 3010 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
21 | 11, 17, 18, 19, 20 | qliftfun 6519 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ⦋csb 3007 〈cop 3535 class class class wbr 3937 ↦ cmpt 3997 ran crn 4548 Fun wfun 5125 Er wer 6434 [cec 6435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-er 6437 df-ec 6439 df-qs 6443 |
This theorem is referenced by: (None) |
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