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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 9341 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1of1 6614 | . . 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 3991 | . 2 ⊢ 𝐴 ⊆ V | |
5 | inlresf 9343 | . 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) | |
6 | f1resf1 6583 | . 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 1457 | 1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3494 ⊆ wss 3936 ∅c0 4291 {csn 4567 × cxp 5553 ↾ cres 5557 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ⊔ cdju 9327 inlcinl 9328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-2nd 7690 df-dju 9330 df-inl 9331 |
This theorem is referenced by: (None) |
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