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Mirrors > Home > ILE Home > Th. List > fvelimab | Unicode version |
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.) |
Ref | Expression |
---|---|
fvelimab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . . . 4 | |
2 | 1 | anim2i 340 | . . 3 |
3 | ssel2 3142 | . . . . . . . 8 | |
4 | funfvex 5511 | . . . . . . . . 9 | |
5 | 4 | funfni 5296 | . . . . . . . 8 |
6 | 3, 5 | sylan2 284 | . . . . . . 7 |
7 | 6 | anassrs 398 | . . . . . 6 |
8 | eleq1 2233 | . . . . . 6 | |
9 | 7, 8 | syl5ibcom 154 | . . . . 5 |
10 | 9 | rexlimdva 2587 | . . . 4 |
11 | 10 | imdistani 443 | . . 3 |
12 | eleq1 2233 | . . . . . . 7 | |
13 | eqeq2 2180 | . . . . . . . 8 | |
14 | 13 | rexbidv 2471 | . . . . . . 7 |
15 | 12, 14 | bibi12d 234 | . . . . . 6 |
16 | 15 | imbi2d 229 | . . . . 5 |
17 | fnfun 5293 | . . . . . . . 8 | |
18 | 17 | adantr 274 | . . . . . . 7 |
19 | fndm 5295 | . . . . . . . . 9 | |
20 | 19 | sseq2d 3177 | . . . . . . . 8 |
21 | 20 | biimpar 295 | . . . . . . 7 |
22 | dfimafn 5543 | . . . . . . 7 | |
23 | 18, 21, 22 | syl2anc 409 | . . . . . 6 |
24 | 23 | abeq2d 2283 | . . . . 5 |
25 | 16, 24 | vtoclg 2790 | . . . 4 |
26 | 25 | impcom 124 | . . 3 |
27 | 2, 11, 26 | pm5.21nd 911 | . 2 |
28 | fveq2 5494 | . . . 4 | |
29 | 28 | eqeq1d 2179 | . . 3 |
30 | 29 | cbvrexv 2697 | . 2 |
31 | 27, 30 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 wrex 2449 cvv 2730 wss 3121 cdm 4609 cima 4612 wfun 5190 wfn 5191 cfv 5196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 |
This theorem is referenced by: ssimaex 5555 foima2 5728 rexima 5731 ralima 5732 f1elima 5749 ovelimab 6000 |
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