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Mirrors > Home > ILE Home > Th. List > eninl | GIF version |
Description: Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
Ref | Expression |
---|---|
eninl | ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 7054 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | f1oeng 6756 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)) → 𝐴 ≈ ({∅} × 𝐴)) | |
3 | 1, 2 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ ({∅} × 𝐴)) |
4 | df-ima 4639 | . . . 4 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴) | |
5 | dff1o5 5470 | . . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴–1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴))) | |
6 | 1, 5 | mpbi 145 | . . . . 5 ⊢ ((inl ↾ 𝐴):𝐴–1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴)) |
7 | 6 | simpri 113 | . . . 4 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴) |
8 | 4, 7 | eqtri 2198 | . . 3 ⊢ (inl “ 𝐴) = ({∅} × 𝐴) |
9 | 3, 8 | breqtrrdi 4045 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (inl “ 𝐴)) |
10 | 9 | ensymd 6782 | 1 ⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∅c0 3422 {csn 3592 class class class wbr 4003 × cxp 4624 ran crn 4627 ↾ cres 4628 “ cima 4629 –1-1→wf1 5213 –1-1-onto→wf1o 5215 ≈ cen 6737 inlcinl 7043 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1st 6140 df-2nd 6141 df-er 6534 df-en 6740 df-inl 7045 |
This theorem is referenced by: endjudisj 7208 djuen 7209 |
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