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Mirrors > Home > ILE Home > Th. List > fliftfuns | GIF version |
Description: The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) |
flift.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) |
flift.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
fliftfuns | ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
2 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑦〈𝐴, 𝐵〉 | |
3 | nfcsb1v 2882 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
4 | nfcsb1v 2882 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | 3, 4 | nfop 3565 | . . . . 5 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉 |
6 | csbeq1a 2860 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
7 | csbeq1a 2860 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
8 | 6, 7 | opeq12d 3557 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝐴, 𝐵〉 = 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
9 | 2, 5, 8 | cbvmpt 3851 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
10 | 9 | rneqi 4562 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = ran (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
11 | 1, 10 | eqtri 2060 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
12 | flift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
13 | 12 | ralrimiva 2392 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅) |
14 | 3 | nfel1 2188 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅 |
15 | 6 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑅 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)) |
16 | 14, 15 | rspc 2650 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)) |
17 | 13, 16 | mpan9 265 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅) |
18 | flift.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
19 | 18 | ralrimiva 2392 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆) |
20 | 4 | nfel1 2188 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆 |
21 | 7 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑆 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)) |
22 | 20, 21 | rspc 2650 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)) |
23 | 19, 22 | mpan9 265 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆) |
24 | csbeq1 2855 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
25 | csbeq1 2855 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
26 | 11, 17, 23, 24, 25 | fliftfun 5436 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ⦋csb 2852 〈cop 3378 ↦ cmpt 3818 ran crn 4346 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 |
This theorem is referenced by: (None) |
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