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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 9955 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1ofun 6845 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl) | |
3 | ffvresb 7139 | . . 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 3482 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V) | |
6 | opex 5470 | . . . . 5 ⊢ 〈∅, 𝑥〉 ∈ V | |
7 | df-inl 9945 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | dmmpti 6705 | . . . 4 ⊢ dom inl = V |
9 | 5, 8 | eleqtrrdi 2837 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl) |
10 | djulcl 9953 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 510 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3058 | 1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∅c0 4325 {csn 4633 〈cop 4639 × cxp 5680 dom cdm 5682 ↾ cres 5684 Fun wfun 6548 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 ⊔ cdju 9941 inlcinl 9942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-1st 8003 df-2nd 8004 df-dju 9944 df-inl 9945 |
This theorem is referenced by: inlresf1 9958 updjudhcoinlf 9975 updjud 9977 |
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