Theorem mof0ALT 49768

Description: Alternate proof of mof0 49766 with stronger requirements on distinct variables. Uses mo4 2591. (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 6757	. . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
2	1	simplbi 502	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3		f00 6757	. . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
4	3	simplbi 502	. . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5		eqtr3 2782	. . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
6	2, 4, 5	syl2an 608	. . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
7	6	gen2 1829	. 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8		feq1 6680	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
9	8	mo4 2591	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔))
10	7, 9	mpbir 234	1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃*wmo 2562  ∅c0 4279  ⟶wf 6529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6535  df-fn 6536  df-f 6537

This theorem is used by: (None)