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Theorem brenOLD 8981
Description: Obsolete version of bren 8980 as of 23-Sep-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brenOLD (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem brenOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 encv 8978 . 2 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2 f1ofn 6845 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 Fn 𝐴)
3 fndm 6662 . . . . . 6 (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4 vex 3477 . . . . . . 7 𝑓 ∈ V
54dmex 7923 . . . . . 6 dom 𝑓 ∈ V
63, 5eqeltrrdi 2838 . . . . 5 (𝑓 Fn 𝐴𝐴 ∈ V)
72, 6syl 17 . . . 4 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
8 f1ofo 6851 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
9 forn 6819 . . . . . 6 (𝑓:𝐴onto𝐵 → ran 𝑓 = 𝐵)
108, 9syl 17 . . . . 5 (𝑓:𝐴1-1-onto𝐵 → ran 𝑓 = 𝐵)
114rnex 7924 . . . . 5 ran 𝑓 ∈ V
1210, 11eqeltrrdi 2838 . . . 4 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
137, 12jca 510 . . 3 (𝑓:𝐴1-1-onto𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1413exlimiv 1925 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15 f1oeq2 6833 . . . 4 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑦𝑓:𝐴1-1-onto𝑦))
1615exbidv 1916 . . 3 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑦))
17 f1oeq3 6834 . . . 4 (𝑦 = 𝐵 → (𝑓:𝐴1-1-onto𝑦𝑓:𝐴1-1-onto𝐵))
1817exbidv 1916 . . 3 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
19 df-en 8971 . . 3 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
2016, 18, 19brabg 5545 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
211, 14, 20pm5.21nii 377 1 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473   class class class wbr 5152  dom cdm 5682  ran crn 5683   Fn wfn 6548  ontowfo 6551  1-1-ontowf1o 6552  cen 8967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-en 8971
This theorem is referenced by: (None)
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