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| Mirrors > Home > MPE Home > Th. List > nffvmpt1 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.) |
| Ref | Expression |
|---|---|
| nffvmpt1 | ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfmpt1 5250 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 3 | 1, 2 | nffv 6916 | 1 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 ↦ cmpt 5225 ‘cfv 6561 |
| 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 |
| 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-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-iota 6514 df-fv 6569 |
| This theorem is referenced by: fvmptt 7036 fmptco 7149 offval2f 7712 offval2 7717 ofrfval2 7718 mptelixpg 8975 dom2lem 9032 cantnflem1 9729 acni2 10086 axcc2 10477 seqof2 14101 rlim2 15532 ello1mpt 15557 o1compt 15623 sumfc 15745 fsum 15756 fsumf1o 15759 sumss 15760 fsumcvg2 15763 fsumadd 15776 isummulc2 15798 fsummulc2 15820 fsumrelem 15843 isumshft 15875 zprod 15973 fprod 15977 prodfc 15981 fprodf1o 15982 fprodmul 15996 fproddiv 15997 iserodd 16873 prdsbas3 17526 prdsdsval2 17529 invfuc 18022 yonedalem4b 18321 gsumdixp 20316 evlslem4 22100 elptr2 23582 ptunimpt 23603 ptcldmpt 23622 ptclsg 23623 txcnp 23628 ptcnplem 23629 cnmpt1t 23673 cnmptk2 23694 flfcnp2 24015 voliun 25589 mbfeqalem1 25676 mbfpos 25686 mbfposb 25688 mbfsup 25699 mbfinf 25700 mbflim 25703 i1fposd 25742 isibl2 25801 itgmpt 25818 itgeqa 25849 itggt0 25879 itgcn 25880 limcmpt 25918 lhop2 26054 itgsubstlem 26089 itgsubst 26090 elplyd 26241 coeeq2 26281 dgrle 26282 ulmss 26440 itgulm2 26452 leibpi 26985 rlimcnp 27008 o1cxp 27018 lgamgulmlem2 27073 lgamgulmlem6 27077 fmptcof2 32667 itggt0cn 37697 elrfirn2 42707 eq0rabdioph 42787 monotoddzz 42955 aomclem8 43073 fmuldfeq 45598 vonioo 46697 |
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