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Mirrors > Home > MPE Home > Th. List > fornex | Structured version Visualization version Unicode version |
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
fornex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 5794 |
. . . 4
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2 | funrnex 6760 |
. . . 4
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3 | 1, 2 | syl5com 31 |
. . 3
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4 | fof 5793 |
. . . . 5
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5 | fdm 5733 |
. . . . 5
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6 | 4, 5 | syl 17 |
. . . 4
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7 | 6 | eleq1d 2513 |
. . 3
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8 | forn 5796 |
. . . 4
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9 | 8 | eleq1d 2513 |
. . 3
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10 | 3, 7, 9 | 3imtr3d 271 |
. 2
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11 | 10 | com12 32 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 |
This theorem is referenced by: f1dmex 6763 f1ovv 6764 f1oeng 7588 fodomnum 8488 ttukeylem1 8939 fodomb 8954 cnexALT 11298 imasbas 15413 imasds 15414 imasbasOLD 15425 imasdsOLD 15426 elqtop 20712 qtoprest 20732 indishmph 20813 imasf1oxmet 21390 ghgrpOLD 26096 foresf1o 28139 noprc 30570 bj-finsumval0 31702 sge0f1o 38224 sge0fodjrnlem 38258 |
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