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Theorem brdomg 8746

Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) (Proof shortened by BTernaryTau, 29-Nov-2024.)

**Assertion**
Ref	Expression
brdomg	⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓 Allowed substitution hint: 𝐶(𝑓)

**Proof of Theorem brdomg**
Step	Hyp	Ref	Expression
1		brdom2g 8745	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
2	1	ex 413	. 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
3		reldom 8739	. . . . 5 ⊢ Rel ≼
4	3	brrelex1i 5643	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
5		f1f 6670	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
6		fdm 6609	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
7		vex 3436	. . . . . . . 8 ⊢ 𝑓 ∈ V
8	7	dmex 7758	. . . . . . 7 ⊢ dom 𝑓 ∈ V
9	6, 8	eqeltrrdi 2848	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
10	5, 9	syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
11	10	exlimiv 1933	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
12	4, 11	pm5.21ni 379	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
13	12	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
14	2, 13	pm2.61i 182	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 –1-1→wf1 6430 ≼ cdom 8731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-fn 6436 df-f 6437 df-f1 6438 df-dom 8735

This theorem is referenced by: brdomiOLD 8749 brdom 8750 f1dom3g 8755 f1dom2gOLD 8758 f1domg 8760 dom3d 8782 domdifsn 8841 fidomtri 9751 hashdom 14094 hashge3el3dif 14200 sizusglecusg 27830 erdsze2lem1 33165