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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 9344 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1ofun 6620 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl) | |
3 | ffvresb 6891 | . . 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 3515 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V) | |
6 | opex 5359 | . . . . 5 ⊢ 〈∅, 𝑥〉 ∈ V | |
7 | df-inl 9334 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | dmmpti 6495 | . . . 4 ⊢ dom inl = V |
9 | 5, 8 | eleqtrrdi 2927 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl) |
10 | djulcl 9342 | . . 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 3156 | 1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ∅c0 4294 {csn 4570 〈cop 4576 × cxp 5556 dom cdm 5558 ↾ cres 5560 Fun wfun 6352 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 ⊔ cdju 9330 inlcinl 9331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1st 7692 df-2nd 7693 df-dju 9333 df-inl 9334 |
This theorem is referenced by: inlresf1 9347 updjudhcoinlf 9364 updjud 9366 |
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