Theorem mof0ALT 49618

Description: Alternate proof of mof0 49616 with stronger requirements on distinct variables. Uses mo4 2594. (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 6760	. . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
2	1	simplbi 501	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3		f00 6760	. . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
4	3	simplbi 501	. . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5		eqtr3 2785	. . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
6	2, 4, 5	syl2an 607	. . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
7	6	gen2 1826	. 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8		feq1 6683	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
9	8	mo4 2594	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔))
10	7, 9	mpbir 234	1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃*wmo 2565  ∅c0 4286  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by: (None)