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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 8936 | . . 3 ⊢ inl:V–1-1-onto→({∅} × V) | |
2 | f1ofun 6278 | . . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl) | |
3 | ffvresb 6534 | . . 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 3364 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V) | |
6 | opex 5060 | . . . . 5 ⊢ 〈∅, 𝑥〉 ∈ V | |
7 | df-inl 8927 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | dmmpti 6161 | . . . 4 ⊢ dom inl = V |
9 | 5, 8 | syl6eleqr 2861 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl) |
10 | djulcl 8934 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 501 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3076 | 1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 ∅c0 4063 {csn 4316 〈cop 4322 × cxp 5247 dom cdm 5249 ↾ cres 5251 Fun wfun 6023 ⟶wf 6025 –1-1-onto→wf1o 6028 ‘cfv 6029 ⊔ cdju 8923 inlcinl 8924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-1st 7313 df-2nd 7314 df-dju 8926 df-inl 8927 |
This theorem is referenced by: inlresf1 8939 updjudhcoinlf 8956 updjud 8958 |
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