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Mirrors > Home > MPE Home > Th. List > inrresf | Structured version Visualization version GIF version |
Description: The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
inrresf | ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djurf1o 9951 | . . 3 ⊢ inr:V–1-1-onto→({1o} × V) | |
2 | f1ofun 6851 | . . 3 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr) | |
3 | ffvresb 7145 | . . 3 ⊢ (Fun inr → ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
5 | elex 3499 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ V) | |
6 | opex 5475 | . . . . 5 ⊢ 〈1o, 𝑥〉 ∈ V | |
7 | df-inr 9941 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | dmmpti 6713 | . . . 4 ⊢ dom inr = V |
9 | 5, 8 | eleqtrrdi 2850 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ dom inr) |
10 | djurcl 9949 | . . 3 ⊢ (𝑥 ∈ 𝐵 → (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 511 | . 2 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3066 | 1 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 {csn 4631 〈cop 4637 × cxp 5687 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 1oc1o 8498 ⊔ cdju 9936 inrcinr 9938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-dju 9939 df-inr 9941 |
This theorem is referenced by: inrresf1 9955 updjudhcoinrg 9971 updjud 9972 |
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