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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 7053 | . . . 4 ⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴) | |
2 | f1oeng 6754 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴)) → 𝐴 ≈ ({1o} × 𝐴)) | |
3 | 1, 2 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ ({1o} × 𝐴)) |
4 | df-ima 4638 | . . . 4 ⊢ (inr “ 𝐴) = ran (inr ↾ 𝐴) | |
5 | dff1o5 5469 | . . . . . 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 145 | . . . . 5 ⊢ ((inr ↾ 𝐴):𝐴–1-1→({1o} × 𝐴) ∧ ran (inr ↾ 𝐴) = ({1o} × 𝐴)) |
7 | 6 | simpri 113 | . . . 4 ⊢ ran (inr ↾ 𝐴) = ({1o} × 𝐴) |
8 | 4, 7 | eqtri 2198 | . . 3 ⊢ (inr “ 𝐴) = ({1o} × 𝐴) |
9 | 3, 8 | breqtrrdi 4044 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (inr “ 𝐴)) |
10 | 9 | ensymd 6780 | 1 ⊢ (𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {csn 3592 class class class wbr 4002 × cxp 4623 ran crn 4626 ↾ cres 4627 “ cima 4628 –1-1→wf1 5212 –1-1-onto→wf1o 5214 1oc1o 6407 ≈ cen 6735 inrcinr 7042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-suc 4370 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-1st 6138 df-2nd 6139 df-1o 6414 df-er 6532 df-en 6738 df-inr 7044 |
This theorem is referenced by: endjudisj 7206 djuen 7207 |
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