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 2381. See cbvmptfg 5157 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 1906 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) | |
2 | cbvmptf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2968 | . . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
4 | nfs1v 2264 | . . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵 | |
5 | 3, 4 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
6 | eleq1w 2892 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) | |
7 | sbequ12 2243 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵)) | |
8 | 6, 7 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵))) |
9 | 1, 5, 8 | cbvopab1 5130 | . . 3 ⊢ {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} |
10 | cbvmptf.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
11 | 10 | nfcri 2968 | . . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴 |
12 | cbvmptf.3 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
13 | 12 | nfeq2 2992 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵 |
14 | 13 | nfsbv 2340 | . . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵 |
15 | 11, 14 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
16 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶) | |
17 | eleq1w 2892 | . . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | sbequ 2081 | . . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵)) | |
19 | cbvmptf.4 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐶 | |
20 | 19 | nfeq2 2992 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶 |
21 | cbvmptf.5 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
22 | 21 | eqeq2d 2829 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
23 | 20, 22 | sbiev 2321 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶) |
24 | 18, 23 | syl6bb 288 | . . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
25 | 17, 24 | anbi12d 630 | . . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶))) |
26 | 15, 16, 25 | cbvopab1 5130 | . . 3 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
27 | 9, 26 | eqtri 2841 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
28 | df-mpt 5138 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
29 | df-mpt 5138 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} | |
30 | 27, 28, 29 | 3eqtr4i 2851 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 [wsb 2060 ∈ wcel 2105 Ⅎwnfc 2958 {copab 5119 ↦ cmpt 5137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-mpt 5138 |
This theorem is referenced by: cbvmpt 5158 resmptf 5900 fvmpt2f 6762 offval2f 7410 suppss2f 30312 fmptdF 30329 acunirnmpt2f 30334 funcnv4mpt 30342 cbvesum 31200 esumpfinvalf 31234 binomcxplemdvbinom 40562 binomcxplemdvsum 40564 binomcxplemnotnn0 40565 supxrleubrnmptf 41603 fnlimfv 41820 fnlimfvre2 41834 fnlimf 41835 limsupequzmptf 41888 sge0iunmptlemre 42574 smflim 42930 smflim2 42957 smfsup 42965 smfinf 42969 smflimsuplem2 42972 smflimsuplem5 42975 smflimsup 42979 smfliminf 42982 |
Copyright terms: Public domain | W3C validator |