Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2907 | . . . . 5 ⊢ Ⅎ𝑦〈[𝑥]𝑅, 𝐴〉 | |
3 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅 | |
4 | nfcsb1v 3856 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfop 4820 | . . . . 5 ⊢ Ⅎ𝑥〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉 |
6 | eceq1 8523 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
7 | csbeq1a 3845 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
8 | 6, 7 | opeq12d 4812 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈[𝑥]𝑅, 𝐴〉 = 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
9 | 2, 5, 8 | cbvmpt 5184 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
10 | 9 | rneqi 5839 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
11 | 1, 10 | eqtri 2766 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴〉) |
12 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
13 | 12 | ralrimiva 3108 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
14 | 4 | nfel1 2923 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌 |
15 | 7 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
16 | 14, 15 | rspc 3546 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)) |
17 | 13, 16 | mpan9 507 | . 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 3834 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
21 | 11, 17, 18, 19, 20 | qliftfun 8578 | 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 396 ∀wal 1537 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⦋csb 3831 〈cop 4567 class class class wbr 5073 ↦ cmpt 5156 ran crn 5585 Fun wfun 6420 Er wer 8482 [cec 8483 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fv 6434 df-er 8485 df-ec 8487 df-qs 8491 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |