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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 5051 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | nfima.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfima.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfres 5319 | . . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
5 | 4 | nfrn 5289 | . 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵) |
6 | 1, 5 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 ran crn 5039 ↾ cres 5040 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: nfimad 5394 csbima12 5402 nfpred 5602 nfsup 8240 nfoi 8302 nfseq 12673 gsum2d2 18196 ptbasfi 21194 mbfposr 23225 itg1climres 23287 limciun 23464 funimass4f 28818 poimirlem16 32595 poimirlem19 32598 aomclem8 36649 areaquad 36821 binomcxplemdvbinom 37574 binomcxplemdvsum 37576 binomcxplemnotnn0 37577 rfcnpre1 38201 rfcnpre2 38213 |
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