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| Mirrors > Home > MPE Home > Th. List > nfima | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| nfima.1 | ⊢ Ⅎ𝑥𝐴 |
| nfima.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| nfima | ⊢ Ⅎ𝑥(𝐴 “ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ima 5698 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
| 2 | nfima.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfima.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 4 | 2, 3 | nfres 5999 | . . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
| 5 | 4 | nfrn 5963 | . 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵) |
| 6 | 1, 5 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| 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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: nfimad 6087 csbima12 6097 nfpred 6326 nfsup 9491 nfoi 9554 nfseq 14052 gsum2d2 19992 ptbasfi 23589 mbfposr 25687 itg1climres 25749 limciun 25929 nfseqs 28293 funimass4f 32647 poimirlem16 37643 poimirlem19 37646 aomclem8 43073 areaquad 43228 nfcoll 44275 binomcxplemdvbinom 44372 binomcxplemdvsum 44374 binomcxplemnotnn0 44375 rfcnpre1 45024 rfcnpre2 45036 smfpimcc 46823 |
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