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Mirrors > Home > MPE Home > Th. List > inlresf | Structured version Visualization version GIF version |
Description: The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
inlresf | ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1o 9341 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1ofun 6617 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl) | |
3 | ffvresb 6888 | . . 3 ⊢ (Fun inl → ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
5 | elex 3512 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V) | |
6 | opex 5356 | . . . . 5 ⊢ 〈∅, 𝑥〉 ∈ V | |
7 | df-inl 9331 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | dmmpti 6492 | . . . 4 ⊢ dom inl = V |
9 | 5, 8 | eleqtrrdi 2924 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl) |
10 | djulcl 9339 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 514 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3153 | 1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ∅c0 4291 {csn 4567 〈cop 4573 × cxp 5553 dom cdm 5555 ↾ cres 5557 Fun wfun 6349 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 ⊔ 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: inlresf1 9344 updjudhcoinlf 9361 updjud 9363 |
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