![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fliftfuns | Structured version Visualization version 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 2751 | . . . . 5 ⊢ Ⅎ𝑦〈𝐴, 𝐵〉 | |
3 | nfcsb1v 3515 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
4 | nfcsb1v 3515 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | 3, 4 | nfop 4356 | . . . . 5 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉 |
6 | csbeq1a 3508 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
7 | csbeq1a 3508 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
8 | 6, 7 | opeq12d 4348 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝐴, 𝐵〉 = 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
9 | 2, 5, 8 | cbvmpt 4677 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
10 | 9 | rneqi 5273 | . . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = ran (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
11 | 1, 10 | eqtri 2632 | . 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ 〈⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵〉) |
12 | flift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
13 | 12 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅) |
14 | 3 | nfel1 2765 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅 |
15 | 6 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑅 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)) |
16 | 14, 15 | rspc 3276 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)) |
17 | 13, 16 | mpan9 485 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅) |
18 | flift.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
19 | 18 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆) |
20 | 4 | nfel1 2765 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆 |
21 | 7 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑆 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)) |
22 | 20, 21 | rspc 3276 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)) |
23 | 19, 22 | mpan9 485 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆) |
24 | csbeq1 3502 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
25 | csbeq1 3502 | . 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
26 | 11, 17, 23, 24, 25 | fliftfun 6462 | 1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⦋csb 3499 〈cop 4131 ↦ cmpt 4643 ran crn 5039 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |