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Mirrors > Home > MPE Home > Th. List > inlresf1 | Structured version Visualization version GIF version |
Description: The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
inlresf1 | ⢠(inl ā¾ š“):š“ā1-1ā(š“ ā šµ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1o 9943 | . . 3 ⢠inl:Vā1-1-ontoā({ā } Ć V) | |
2 | f1of1 6843 | . . 3 ⢠(inl:Vā1-1-ontoā({ā } Ć V) ā inl:Vā1-1ā({ā } Ć V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⢠inl:Vā1-1ā({ā } Ć V) |
4 | ssv 4006 | . 2 ⢠š“ ā V | |
5 | inlresf 9945 | . 2 ⢠(inl ā¾ š“):š“ā¶(š“ ā šµ) | |
6 | f1resf1 6807 | . 2 ⢠((inl:Vā1-1ā({ā } Ć V) ā§ š“ ā V ā§ (inl ā¾ š“):š“ā¶(š“ ā šµ)) ā (inl ā¾ š“):š“ā1-1ā(š“ ā šµ)) | |
7 | 3, 4, 5, 6 | mp3an 1457 | 1 ⢠(inl ā¾ š“):š“ā1-1ā(š“ ā šµ) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3473 ā wss 3949 ā c0 4326 {csn 4632 Ć cxp 5680 ā¾ cres 5684 ā¶wf 6549 ā1-1āwf1 6550 ā1-1-ontoāwf1o 6552 ā cdju 9929 inlcinl 9930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-1st 7999 df-2nd 8000 df-dju 9932 df-inl 9933 |
This theorem is referenced by: (None) |
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