Theorem mofeu 50053

Description: The uniqueness of a function into a set with at most one element. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
mofeu.1	⊢ 𝐺 = (𝐴 × 𝐵)
mofeu.2	⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
mofeu.3	⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
mofeu	⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem mofeu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mofeu.2	. . . . 5 ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
2	1	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 = ∅)
3		f00 6765	. . . . 5 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	rbaib 548	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
5	2, 4	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
6		feq3 6690	. . . 4 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
7	6	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
8		mofeu.1	. . . . . 6 ⊢ 𝐺 = (𝐴 × 𝐵)
9		xpeq2 5671	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10		xp0 5750	. . . . . . 7 ⊢ (𝐴 × ∅) = ∅
11	9, 10	eqtrdi 2811	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
12	8, 11	eqtrid 2807	. . . . 5 ⊢ (𝐵 = ∅ → 𝐺 = ∅)
13	12	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐺 = ∅)
14	13	eqeq2d 2771	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹 = 𝐺 ↔ 𝐹 = ∅))
15	5, 7, 14	3bitr4d 314	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
16		19.42v 1985	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17		fconst2g 7210	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
18	17	elv 3455	. . . . . . 7 ⊢ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19		feq3 6690	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶{𝑦}))
20		xpeq2 5671	. . . . . . . . 9 ⊢ (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
21	20	eqeq2d 2771	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹 = (𝐴 × 𝐵) ↔ 𝐹 = (𝐴 × {𝑦})))
22	19, 21	bibi12d 348	. . . . . . 7 ⊢ (𝐵 = {𝑦} → ((𝐹:𝐴⟶𝐵 ↔ 𝐹 = (𝐴 × 𝐵)) ↔ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))))
23	18, 22	mpbiri 261	. . . . . 6 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = (𝐴 × 𝐵)))
24	8	eqeq2i 2773	. . . . . 6 ⊢ (𝐹 = 𝐺 ↔ 𝐹 = (𝐴 × 𝐵))
25	23, 24	bitr4di 292	. . . . 5 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
26	25	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
27	26	exlimiv 1962	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
28	16, 27	sylbir 238	. 2 ⊢ ((𝜑 ∧ ∃𝑦 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
29		mofeu.3	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
30		mo0sn 50021	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
31	29, 30	sylib 221	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
32	15, 28, 31	mpjaodan 973	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  Vcvv 3450  ∅c0 4278  {csn 4584   × cxp 5648  ⟶wf 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547

This theorem is used by:  functhinclem1  50673  functhinclem3  50675