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Mirrors > Home > ILE Home > Th. List > eninr | GIF version |
Description: Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
Ref | Expression |
---|---|
eninr | ⊢ (𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djurf1or 6950 | . . . 4 ⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴) | |
2 | f1oeng 6659 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴)) → 𝐴 ≈ ({1o} × 𝐴)) | |
3 | 1, 2 | mpan2 422 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ ({1o} × 𝐴)) |
4 | df-ima 4560 | . . . 4 ⊢ (inr “ 𝐴) = ran (inr ↾ 𝐴) | |
5 | dff1o5 5384 | . . . . . 6 ⊢ ((inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴) ↔ ((inr ↾ 𝐴):𝐴–1-1→({1o} × 𝐴) ∧ ran (inr ↾ 𝐴) = ({1o} × 𝐴))) | |
6 | 1, 5 | mpbi 144 | . . . . 5 ⊢ ((inr ↾ 𝐴):𝐴–1-1→({1o} × 𝐴) ∧ ran (inr ↾ 𝐴) = ({1o} × 𝐴)) |
7 | 6 | simpri 112 | . . . 4 ⊢ ran (inr ↾ 𝐴) = ({1o} × 𝐴) |
8 | 4, 7 | eqtri 2161 | . . 3 ⊢ (inr “ 𝐴) = ({1o} × 𝐴) |
9 | 3, 8 | breqtrrdi 3978 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (inr “ 𝐴)) |
10 | 9 | ensymd 6685 | 1 ⊢ (𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 {csn 3532 class class class wbr 3937 × cxp 4545 ran crn 4548 ↾ cres 4549 “ cima 4550 –1-1→wf1 5128 –1-1-onto→wf1o 5130 1oc1o 6314 ≈ cen 6640 inrcinr 6939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 df-1o 6321 df-er 6437 df-en 6643 df-inr 6941 |
This theorem is referenced by: endjudisj 7083 djuen 7084 |
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