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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 2371. See cbvmptfg 5219 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 5184 | . . 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 5184 | . . 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 5193 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
29 | df-mpt 5193 | . 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 5171 ↦ cmpt 5192 |
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 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-opab 5172 df-mpt 5193 |
This theorem is referenced by: cbvmpt 5220 resmptf 5997 fvmpt2f 6953 offval2f 7636 suppss2f 31606 fmptdF 31625 acunirnmpt2f 31630 funcnv4mpt 31638 cbvesum 32705 esumpfinvalf 32739 binomcxplemdvbinom 42725 binomcxplemdvsum 42727 binomcxplemnotnn0 42728 fmptff 43589 supxrleubrnmptf 43776 fnlimfv 43994 fnlimfvre2 44008 fnlimf 44009 limsupequzmptf 44062 sge0iunmptlemre 44746 smflim 45108 smflim2 45137 smfsup 45145 smfinf 45149 smflimsuplem2 45152 smflimsuplem5 45155 smflimsup 45159 smfliminf 45162 |
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