Theorem mof0ALT 48754

Description: Alternate proof of mof0 48752 with stronger requirements on distinct variables. Uses mo4 2565. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mof0ALT	⊢ ∃*𝑓 𝑓:𝐴⟶∅

Distinct variable group:   𝐴,𝑓

Proof of Theorem mof0ALT

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f00 6789	. . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
2	1	simplbi 497	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3		f00 6789	. . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
4	3	simplbi 497	. . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5		eqtr3 2762	. . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
6	2, 4, 5	syl2an 596	. . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
7	6	gen2 1795	. 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8		feq1 6715	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
9	8	mo4 2565	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔))
10	7, 9	mpbir 231	1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃*wmo 2537  ∅c0 4332  ⟶wf 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-f 6564

This theorem is referenced by: (None)