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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 6749 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | f1ofn 5464 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 5317 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 2742 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 4895 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2269 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V) |
8 | f1ofo 5470 | . . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
9 | forn 5443 | . . . . . 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 4896 | . . . . 5 ⊢ ran 𝑓 ∈ V |
12 | 10, 11 | eqeltrrdi 2269 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V) |
13 | 7, 12 | jca 306 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
14 | 13 | exlimiv 1598 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | f1oeq2 5452 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦)) | |
16 | 15 | exbidv 1825 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦)) |
17 | f1oeq3 5453 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵)) | |
18 | 17 | exbidv 1825 | . . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
19 | df-en 6744 | . . 3 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
20 | 16, 18, 19 | brabg 4271 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
21 | 1, 14, 20 | pm5.21nii 704 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2739 class class class wbr 4005 dom cdm 4628 ran crn 4629 Fn wfn 5213 –onto→wfo 5216 –1-1-onto→wf1o 5217 ≈ cen 6741 |
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-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-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
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-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-rel 4635 df-cnv 4636 df-dm 4638 df-rn 4639 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-en 6744 |
This theorem is referenced by: domen 6754 f1oen3g 6757 ener 6782 en0 6798 ensn1 6799 en1 6802 unen 6819 enm 6823 xpen 6848 mapen 6849 ssenen 6854 phplem4 6858 phplem4on 6870 fidceq 6872 dif1en 6882 fin0 6888 fin0or 6889 en2eqpr 6910 fiintim 6931 fidcenumlemim 6954 enomnilem 7139 enmkvlem 7162 enwomnilem 7170 cc3 7270 hasheqf1o 10768 hashfacen 10819 fz1f1o 11386 eulerth 12236 ennnfonelemim 12428 exmidunben 12430 ctinfom 12432 qnnen 12435 enctlem 12436 ctiunct 12444 exmidsbthrlem 14932 sbthom 14936 |
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