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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 9493 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1ofun 6641 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl) | |
3 | ffvresb 6919 | . . 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 3416 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V) | |
6 | opex 5333 | . . . . 5 ⊢ 〈∅, 𝑥〉 ∈ V | |
7 | df-inl 9483 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | dmmpti 6500 | . . . 4 ⊢ dom inl = V |
9 | 5, 8 | eleqtrrdi 2842 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl) |
10 | djulcl 9491 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 515 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3066 | 1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ∅c0 4223 {csn 4527 〈cop 4533 × cxp 5534 dom cdm 5536 ↾ cres 5538 Fun wfun 6352 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 ⊔ cdju 9479 inlcinl 9480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1st 7739 df-2nd 7740 df-dju 9482 df-inl 9483 |
This theorem is referenced by: inlresf1 9496 updjudhcoinlf 9513 updjud 9515 |
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