Theorem mofeu 46175

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 407	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 = ∅)
3		f00 6656	. . . . 5 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	rbaib 539	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
5	2, 4	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
6		feq3 6583	. . . 4 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
7	6	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
8		mofeu.1	. . . . . 6 ⊢ 𝐺 = (𝐴 × 𝐵)
9		xpeq2 5610	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10		xp0 6061	. . . . . . 7 ⊢ (𝐴 × ∅) = ∅
11	9, 10	eqtrdi 2794	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
12	8, 11	eqtrid 2790	. . . . 5 ⊢ (𝐵 = ∅ → 𝐺 = ∅)
13	12	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐺 = ∅)
14	13	eqeq2d 2749	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹 = 𝐺 ↔ 𝐹 = ∅))
15	5, 7, 14	3bitr4d 311	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
16		19.42v 1957	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17		fconst2g 7078	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
18	17	elv 3438	. . . . . . 7 ⊢ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19		feq3 6583	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶{𝑦}))
20		xpeq2 5610	. . . . . . . . 9 ⊢ (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
21	20	eqeq2d 2749	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹 = (𝐴 × 𝐵) ↔ 𝐹 = (𝐴 × {𝑦})))
22	19, 21	bibi12d 346	. . . . . . 7 ⊢ (𝐵 = {𝑦} → ((𝐹:𝐴⟶𝐵 ↔ 𝐹 = (𝐴 × 𝐵)) ↔ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))))
23	18, 22	mpbiri 257	. . . . . 6 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = (𝐴 × 𝐵)))
24	8	eqeq2i 2751	. . . . . 6 ⊢ (𝐹 = 𝐺 ↔ 𝐹 = (𝐴 × 𝐵))
25	23, 24	bitr4di 289	. . . . 5 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
26	25	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
27	26	exlimiv 1933	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
28	16, 27	sylbir 234	. 2 ⊢ ((𝜑 ∧ ∃𝑦 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
29		mofeu.3	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
30		mo0sn 46161	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
31	29, 30	sylib 217	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
32	15, 28, 31	mpjaodan 956	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃*wmo 2538  Vcvv 3432  ∅c0 4256  {csn 4561   × cxp 5587  ⟶wf 6429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  functhinclem1  46322  functhinclem3  46324