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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 1916 | . 2 ⊢ Ⅎ𝑖𝜑 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑖𝐴 | |
3 | nfcv 2902 | . 2 ⊢ Ⅎ𝑖𝐹 | |
4 | funcnvmpt.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | funcnvmpt.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑖𝐵 | |
7 | nfcsb1v 3919 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
8 | csbeq1a 3908 | . . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
9 | 5, 2, 6, 7, 8 | cbvmptf 5258 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) |
10 | 4, 9 | eqtri 2759 | . 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) |
11 | funcnvmpt.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
12 | 11 | sbimi 2076 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉) |
13 | funcnvmpt.0 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
14 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝑖 | |
15 | 14, 5 | nfel 2916 | . . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴 |
16 | 13, 15 | nfan 1901 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴) |
17 | eleq1w 2815 | . . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) | |
18 | 17 | anbi2d 628 | . . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))) |
19 | 16, 18 | sbiev 2307 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)) |
20 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝑉 | |
21 | 7, 20 | nfel 2916 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉 |
22 | 8 | eleq1d 2817 | . . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)) |
23 | 21, 22 | sbiev 2307 | . . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) |
24 | 12, 19, 23 | 3imtr3i 290 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) |
25 | csbeq1 3897 | . 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
26 | 1, 2, 3, 10, 24, 25 | funcnv5mpt 32157 | 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 205 ∧ wa 395 ∨ wo 844 = wceq 1540 Ⅎwnf 1784 [wsb 2066 ∈ wcel 2105 Ⅎwnfc 2882 ≠ wne 2939 ∀wral 3060 ⦋csb 3894 ↦ cmpt 5232 ◡ccnv 5676 Fun wfun 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 |
This theorem is referenced by: disjdsct 32188 |
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