![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cbvmptf | Structured version Visualization version GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2372. See cbvmptfg 5259 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
Ref | Expression |
---|---|
cbvmptf.1 | ⊢ Ⅎ𝑥𝐴 |
cbvmptf.2 | ⊢ Ⅎ𝑦𝐴 |
cbvmptf.3 | ⊢ Ⅎ𝑦𝐵 |
cbvmptf.4 | ⊢ Ⅎ𝑥𝐶 |
cbvmptf.5 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmptf | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) | |
2 | cbvmptf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2891 | . . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
4 | nfs1v 2154 | . . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵 | |
5 | 3, 4 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
6 | eleq1w 2817 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) | |
7 | sbequ12 2244 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵)) | |
8 | 6, 7 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵))) |
9 | 1, 5, 8 | cbvopab1 5224 | . . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} |
10 | cbvmptf.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
11 | 10 | nfcri 2891 | . . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴 |
12 | cbvmptf.3 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
13 | 12 | nfeq2 2921 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵 |
14 | 13 | nfsbv 2324 | . . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵 |
15 | 11, 14 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
16 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶) | |
17 | eleq1w 2817 | . . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | sbequ 2087 | . . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵)) | |
19 | cbvmptf.4 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐶 | |
20 | 19 | nfeq2 2921 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶 |
21 | cbvmptf.5 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
22 | 21 | eqeq2d 2744 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
23 | 20, 22 | sbiev 2309 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶) |
24 | 18, 23 | bitrdi 287 | . . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
25 | 17, 24 | anbi12d 632 | . . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶))) |
26 | 15, 16, 25 | cbvopab1 5224 | . . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
27 | 9, 26 | eqtri 2761 | . 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
28 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
29 | df-mpt 5233 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} | |
30 | 27, 28, 29 | 3eqtr4i 2771 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 [wsb 2068 ∈ wcel 2107 Ⅎwnfc 2884 {copab 5211 ↦ cmpt 5232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 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-opab 5212 df-mpt 5233 |
This theorem is referenced by: cbvmpt 5260 resmptf 6040 fvmpt2f 7000 offval2f 7685 suppss2f 31863 fmptdF 31881 acunirnmpt2f 31886 funcnv4mpt 31894 cbvesum 33040 esumpfinvalf 33074 binomcxplemdvbinom 43112 binomcxplemdvsum 43114 binomcxplemnotnn0 43115 fmptff 43974 supxrleubrnmptf 44161 fnlimfv 44379 fnlimfvre2 44393 fnlimf 44394 limsupequzmptf 44447 sge0iunmptlemre 45131 smflim 45493 smflim2 45522 smfsup 45530 smfinf 45534 smflimsuplem2 45537 smflimsuplem5 45540 smflimsup 45544 smfliminf 45547 |
Copyright terms: Public domain | W3C validator |