Theorem mof0ALT 49196

Description: Alternate proof of mof0 49194 with stronger requirements on distinct variables. Uses mo4 2567. (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 6724	. . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
2	1	simplbi 496	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3		f00 6724	. . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
4	3	simplbi 496	. . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5		eqtr3 2759	. . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
6	2, 4, 5	syl2an 597	. . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
7	6	gen2 1798	. 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8		feq1 6648	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
9	8	mo4 2567	. 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 1540   = wceq 1542  ∃*wmo 2538  ∅c0 4287  ⟶wf 6496

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by: (None)