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Mirrors > Home > ILE Home > Th. List > f1elima | Unicode version |
Description: Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
f1elima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 5389 | . . . 4 | |
2 | fvelimab 5536 | . . . 4 | |
3 | 1, 2 | sylan 281 | . . 3 |
4 | 3 | 3adant2 1005 | . 2 |
5 | ssel 3131 | . . . . . . . 8 | |
6 | 5 | impac 379 | . . . . . . 7 |
7 | f1fveq 5734 | . . . . . . . . . . . 12 | |
8 | 7 | ancom2s 556 | . . . . . . . . . . 11 |
9 | 8 | biimpd 143 | . . . . . . . . . 10 |
10 | 9 | anassrs 398 | . . . . . . . . 9 |
11 | eleq1 2227 | . . . . . . . . . 10 | |
12 | 11 | biimpcd 158 | . . . . . . . . 9 |
13 | 10, 12 | sylan9 407 | . . . . . . . 8 |
14 | 13 | anasss 397 | . . . . . . 7 |
15 | 6, 14 | sylan2 284 | . . . . . 6 |
16 | 15 | anassrs 398 | . . . . 5 |
17 | 16 | rexlimdva 2581 | . . . 4 |
18 | 17 | 3impa 1183 | . . 3 |
19 | eqid 2164 | . . . 4 | |
20 | fveq2 5480 | . . . . . 6 | |
21 | 20 | eqeq1d 2173 | . . . . 5 |
22 | 21 | rspcev 2825 | . . . 4 |
23 | 19, 22 | mpan2 422 | . . 3 |
24 | 18, 23 | impbid1 141 | . 2 |
25 | 4, 24 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wrex 2443 wss 3111 cima 4601 wfn 5177 wf1 5179 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fv 5190 |
This theorem is referenced by: f1imass 5736 iseqf1olemnab 10413 fprodssdc 11517 ctinfom 12304 ssnnctlemct 12322 |
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