Theorem mof0ALT 49422

Description: Alternate proof of mof0 49420 with stronger requirements on distinct variables. Uses mo4 2592. (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 6741	. . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
2	1	simplbi 500	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3		f00 6741	. . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
4	3	simplbi 500	. . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5		eqtr3 2783	. . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
6	2, 4, 5	syl2an 605	. . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
7	6	gen2 1815	. 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8		feq1 6664	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
9	8	mo4 2592	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔))
10	7, 9	mpbir 233	1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃*wmo 2563  ∅c0 4283  ⟶wf 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-f 6520

This theorem is referenced by: (None)