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Mirrors > Home > MPE Home > Th. List > cbvmpt | 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.) |
Ref | Expression |
---|---|
cbvmpt.1 | ⊢ Ⅎ𝑦𝐵 |
cbvmpt.2 | ⊢ Ⅎ𝑥𝐶 |
cbvmpt.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmpt | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) | |
2 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 | |
3 | nfs1v 2437 | . . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵 | |
4 | 2, 3 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
5 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) | |
6 | sbequ12 2111 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵)) | |
7 | 5, 6 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵))) |
8 | 1, 4, 7 | cbvopab1 4723 | . . 3 ⊢ {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} |
9 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴 | |
10 | cbvmpt.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
11 | 10 | nfeq2 2780 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵 |
12 | 11 | nfsb 2440 | . . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵 |
13 | 9, 12 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) |
14 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶) | |
15 | eleq1 2689 | . . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
16 | sbequ 2376 | . . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵)) | |
17 | cbvmpt.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐶 | |
18 | 17 | nfeq2 2780 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶 |
19 | cbvmpt.3 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
20 | 19 | eqeq2d 2632 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
21 | 18, 20 | sbie 2408 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶) |
22 | 16, 21 | syl6bb 276 | . . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)) |
23 | 15, 22 | anbi12d 747 | . . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶))) |
24 | 13, 14, 23 | cbvopab1 4723 | . . 3 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
25 | 8, 24 | eqtri 2644 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} |
26 | df-mpt 4730 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
27 | df-mpt 4730 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)} | |
28 | 25, 26, 27 | 3eqtr4i 2654 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 [wsb 1880 ∈ wcel 1990 Ⅎwnfc 2751 {copab 4712 ↦ cmpt 4729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-mpt 4730 |
This theorem is referenced by: cbvmptv 4750 dffn5f 6252 fvmpts 6285 fvmpt2i 6290 fvmptex 6294 fmptcof 6397 fmptcos 6398 fliftfuns 6564 offval2 6914 ofmpteq 6916 mpt2curryvald 7396 qliftfuns 7834 axcc2 9259 ac6num 9301 seqof2 12859 summolem2a 14446 zsum 14449 fsumcvg2 14458 fsumrlim 14543 cbvprod 14645 prodmolem2a 14664 zprod 14667 fprod 14671 pcmptdvds 15598 prdsdsval2 16144 gsumconstf 18335 gsummpt1n0 18364 gsum2d2 18373 dprd2d2 18443 gsumdixp 18609 psrass1lem 19377 coe1fzgsumdlem 19671 gsumply1eq 19675 evl1gsumdlem 19720 madugsum 20449 cnmpt1t 21468 cnmpt2k 21491 elmptrab 21630 flfcnp2 21811 prdsxmet 22174 fsumcn 22673 ovoliunlem3 23272 ovoliun 23273 ovoliun2 23274 voliun 23322 mbfpos 23418 mbfposb 23420 i1fposd 23474 itg2cnlem1 23528 isibl2 23533 cbvitg 23542 itgss3 23581 itgfsum 23593 itgabs 23601 itgcn 23609 limcmpt 23647 dvmptfsum 23738 lhop2 23778 dvfsumle 23784 dvfsumlem2 23790 itgsubstlem 23811 itgsubst 23812 itgulm2 24163 rlimcnp2 24693 gsummpt2co 29780 esumsnf 30126 mbfposadd 33457 itgabsnc 33479 ftc1cnnclem 33483 ftc2nc 33494 mzpsubst 37311 rabdiophlem2 37366 aomclem8 37631 fsumcnf 39180 disjf1 39369 disjrnmpt2 39375 disjinfi 39380 fmptf 39448 cncfmptss 39819 mulc1cncfg 39821 expcnfg 39823 fprodcn 39832 fnlimabslt 39911 climmptf 39913 liminfvalxr 40015 icccncfext 40100 cncficcgt0 40101 cncfiooicclem1 40106 fprodcncf 40114 dvmptmulf 40152 iblsplitf 40186 stoweidlem21 40238 stirlinglem4 40294 stirlinglem13 40303 stirlinglem15 40305 fourierd 40439 fourierclimd 40440 sge0iunmptlemre 40632 sge0iunmpt 40635 sge0ltfirpmpt2 40643 sge0isummpt2 40649 sge0xaddlem2 40651 sge0xadd 40652 meadjiun 40683 omeiunle 40731 caratheodorylem2 40741 ovncvrrp 40778 vonioo 40896 smflim2 41012 smfsup 41020 smfinf 41024 smflimsup 41034 smfliminf 41037 |
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