Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcnv4mpt | Structured version Visualization version GIF version |
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
Ref | Expression |
---|---|
funcnvmpt.0 | ⊢ Ⅎ𝑥𝜑 |
funcnvmpt.1 | ⊢ Ⅎ𝑥𝐴 |
funcnvmpt.2 | ⊢ Ⅎ𝑥𝐹 |
funcnvmpt.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
funcnvmpt.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
funcnv4mpt | ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑖𝜑 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑖𝐴 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑖𝐹 | |
4 | funcnvmpt.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | funcnvmpt.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑖𝐵 | |
7 | nfcsb1v 3549 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
8 | csbeq1a 3542 | . . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
9 | 5, 2, 6, 7, 8 | cbvmptf 4748 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) |
10 | 4, 9 | eqtri 2644 | . 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) |
11 | funcnvmpt.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
12 | 11 | sbimi 1886 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉) |
13 | funcnvmpt.0 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
14 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥𝑖 | |
15 | 14, 5 | nfel 2777 | . . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴 |
16 | 13, 15 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴) |
17 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) | |
18 | 17 | anbi2d 740 | . . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))) |
19 | 16, 18 | sbie 2408 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)) |
20 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥𝑉 | |
21 | 7, 20 | nfel 2777 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉 |
22 | 8 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)) |
23 | 21, 22 | sbie 2408 | . . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) |
24 | 12, 19, 23 | 3imtr3i 280 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) |
25 | csbeq1 3536 | . 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
26 | 1, 2, 3, 10, 24, 25 | funcnv5mpt 29469 | 1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 [wsb 1880 ∈ wcel 1990 Ⅎwnfc 2751 ≠ wne 2794 ∀wral 2912 ⦋csb 3533 ↦ cmpt 4729 ◡ccnv 5113 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: disjdsct 29480 |
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