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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 6724 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | f1ofn 5443 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 5297 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 2733 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 4877 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2262 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V) |
8 | f1ofo 5449 | . . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
9 | forn 5423 | . . . . . 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 4878 | . . . . 5 ⊢ ran 𝑓 ∈ V |
12 | 10, 11 | eqeltrrdi 2262 | . . . 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 1591 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | f1oeq2 5432 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦)) | |
16 | 15 | exbidv 1818 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦)) |
17 | f1oeq3 5433 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵)) | |
18 | 17 | exbidv 1818 | . . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
19 | df-en 6719 | . . 3 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
20 | 16, 18, 19 | brabg 4254 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
21 | 1, 14, 20 | pm5.21nii 699 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1348 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 class class class wbr 3989 dom cdm 4611 ran crn 4612 Fn wfn 5193 –onto→wfo 5196 –1-1-onto→wf1o 5197 ≈ cen 6716 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-dm 4621 df-rn 4622 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-en 6719 |
This theorem is referenced by: domen 6729 f1oen3g 6732 ener 6757 en0 6773 ensn1 6774 en1 6777 unen 6794 enm 6798 xpen 6823 mapen 6824 ssenen 6829 phplem4 6833 phplem4on 6845 fidceq 6847 dif1en 6857 fin0 6863 fin0or 6864 en2eqpr 6885 fiintim 6906 fidcenumlemim 6929 enomnilem 7114 enmkvlem 7137 enwomnilem 7145 cc3 7230 hasheqf1o 10719 hashfacen 10771 fz1f1o 11338 eulerth 12187 ennnfonelemim 12379 exmidunben 12381 ctinfom 12383 qnnen 12386 enctlem 12387 ctiunct 12395 exmidsbthrlem 14054 sbthom 14058 |
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