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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 9909 | . . 3 ⢠inl:Vā1-1-ontoā({ā } Ć V) | |
2 | f1of1 6826 | . . 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 4001 | . 2 ⢠š“ ā V | |
5 | inlresf 9911 | . 2 ⢠(inl ā¾ š“):š“ā¶(š“ ā šµ) | |
6 | f1resf1 6790 | . 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 3468 ā wss 3943 ā c0 4317 {csn 4623 Ć cxp 5667 ā¾ cres 5671 ā¶wf 6533 ā1-1āwf1 6534 ā1-1-ontoāwf1o 6536 ā cdju 9895 inlcinl 9896 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1st 7974 df-2nd 7975 df-dju 9898 df-inl 9899 |
This theorem is referenced by: (None) |
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