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Theorem brdomgOLD 8855
Description: Obsolete version of brdomg 8854 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brdomgOLD (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem brdomgOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq2 6731 . . . . 5 (𝑥 = 𝐴 → (𝑓:𝑥1-1𝑦𝑓:𝐴1-1𝑦))
21exbidv 1924 . . . 4 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦))
3 f1eq3 6732 . . . . 5 (𝑦 = 𝐵 → (𝑓:𝐴1-1𝑦𝑓:𝐴1-1𝐵))
43exbidv 1924 . . . 4 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
5 df-dom 8843 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
62, 4, 5brabg 5494 . . 3 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
76ex 413 . 2 (𝐴 ∈ V → (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)))
8 reldom 8847 . . . . 5 Rel ≼
98brrelex1i 5686 . . . 4 (𝐴𝐵𝐴 ∈ V)
10 f1f 6735 . . . . . 6 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
11 fdm 6674 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
12 vex 3447 . . . . . . . 8 𝑓 ∈ V
1312dmex 7840 . . . . . . 7 dom 𝑓 ∈ V
1411, 13eqeltrrdi 2847 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
1510, 14syl 17 . . . . 5 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
1615exlimiv 1933 . . . 4 (∃𝑓 𝑓:𝐴1-1𝐵𝐴 ∈ V)
179, 16pm5.21ni 378 . . 3 𝐴 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
1817a1d 25 . 2 𝐴 ∈ V → (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)))
197, 18pm2.61i 182 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wex 1781  wcel 2106  Vcvv 3443   class class class wbr 5103  dom cdm 5631  wf 6489  1-1wf1 6490  cdom 8839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-fn 6496  df-f 6497  df-f1 6498  df-dom 8843
This theorem is referenced by: (None)
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