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Mirrors > Home > ILE Home > Th. List > bren | GIF version |
Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
bren | ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 6633 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | f1ofn 5361 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 5217 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 2684 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 4800 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2229 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V) |
8 | f1ofo 5367 | . . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
9 | forn 5343 | . . . . . 6 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵) | |
10 | 8, 9 | syl 14 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ran 𝑓 = 𝐵) |
11 | 4 | rnex 4801 | . . . . 5 ⊢ ran 𝑓 ∈ V |
12 | 10, 11 | eqeltrrdi 2229 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V) |
13 | 7, 12 | jca 304 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
14 | 13 | exlimiv 1577 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | f1oeq2 5352 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦)) | |
16 | 15 | exbidv 1797 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦)) |
17 | f1oeq3 5353 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵)) | |
18 | 17 | exbidv 1797 | . . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
19 | df-en 6628 | . . 3 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
20 | 16, 18, 19 | brabg 4186 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
21 | 1, 14, 20 | pm5.21nii 693 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 class class class wbr 3924 dom cdm 4534 ran crn 4535 Fn wfn 5113 –onto→wfo 5116 –1-1-onto→wf1o 5117 ≈ cen 6625 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-en 6628 |
This theorem is referenced by: domen 6638 f1oen3g 6641 ener 6666 en0 6682 ensn1 6683 en1 6686 unen 6703 enm 6707 xpen 6732 mapen 6733 ssenen 6738 phplem4 6742 phplem4on 6754 fidceq 6756 dif1en 6766 fin0 6772 fin0or 6773 en2eqpr 6794 fiintim 6810 fidcenumlemim 6833 enomnilem 7003 hasheqf1o 10524 hashfacen 10572 fz1f1o 11137 ennnfonelemim 11926 exmidunben 11928 ctinfom 11930 qnnen 11933 enctlem 11934 ctiunct 11942 exmidsbthrlem 13206 sbthom 13210 |
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