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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 ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1o 9187 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1of1 6482 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ inl:V–1-1→({∅} × V) |
4 | ssv 3912 | . 2 ⊢ 𝐴 ⊆ V | |
5 | inlresf 9189 | . 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) | |
6 | f1resf1 6451 | . 2 ⊢ ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V ∧ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)) → (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)) | |
7 | 3, 4, 5, 6 | mp3an 1453 | 1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3437 ⊆ wss 3859 ∅c0 4211 {csn 4472 × cxp 5441 ↾ cres 5445 ⟶wf 6221 –1-1→wf1 6222 –1-1-onto→wf1o 6224 ⊔ cdju 9173 inlcinl 9174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-1st 7545 df-2nd 7546 df-dju 9176 df-inl 9177 |
This theorem is referenced by: (None) |
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