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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 1914 | . 2 ⊢ Ⅎ𝑖𝜑 | |
| 2 | nfcv 2905 | . 2 ⊢ Ⅎ𝑖𝐴 | |
| 3 | nfcv 2905 | . 2 ⊢ Ⅎ𝑖𝐹 | |
| 4 | funcnvmpt.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 5 | funcnvmpt.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 6 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑖𝐵 | |
| 7 | nfcsb1v 3923 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
| 8 | csbeq1a 3913 | . . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
| 9 | 5, 2, 6, 7, 8 | cbvmptf 5251 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) | 
| 10 | 4, 9 | eqtri 2765 | . 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵) | 
| 11 | funcnvmpt.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
| 12 | 11 | sbimi 2074 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉) | 
| 13 | funcnvmpt.0 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 14 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝑖 | |
| 15 | 14, 5 | nfel 2920 | . . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴 | 
| 16 | 13, 15 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴) | 
| 17 | eleq1w 2824 | . . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) | |
| 18 | 17 | anbi2d 630 | . . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))) | 
| 19 | 16, 18 | sbiev 2314 | . . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)) | 
| 20 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑉 | |
| 21 | 7, 20 | nfel 2920 | . . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉 | 
| 22 | 8 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)) | 
| 23 | 21, 22 | sbiev 2314 | . . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) | 
| 24 | 12, 19, 23 | 3imtr3i 291 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉) | 
| 25 | csbeq1 3902 | . 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
| 26 | 1, 2, 3, 10, 24, 25 | funcnv5mpt 32678 | 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 206 ∧ wa 395 ∨ wo 848 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ⦋csb 3899 ↦ cmpt 5225 ◡ccnv 5684 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 | 
| This theorem is referenced by: disjdsct 32712 | 
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