Theorem mofeu 47467

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 6770	. . . . 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 6697	. . . 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 5696	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10		xp0 6154	. . . . . . 7 ⊢ (𝐴 × ∅) = ∅
11	9, 10	eqtrdi 2788	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
12	8, 11	eqtrid 2784	. . . . 5 ⊢ (𝐵 = ∅ → 𝐺 = ∅)
13	12	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐺 = ∅)
14	13	eqeq2d 2743	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹 = 𝐺 ↔ 𝐹 = ∅))
15	5, 7, 14	3bitr4d 310	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
16		19.42v 1957	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17		fconst2g 7200	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
18	17	elv 3480	. . . . . . 7 ⊢ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19		feq3 6697	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶{𝑦}))
20		xpeq2 5696	. . . . . . . . 9 ⊢ (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
21	20	eqeq2d 2743	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹 = (𝐴 × 𝐵) ↔ 𝐹 = (𝐴 × {𝑦})))
22	19, 21	bibi12d 345	. . . . . . 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 2745	. . . . . 6 ⊢ (𝐹 = 𝐺 ↔ 𝐹 = (𝐴 × 𝐵))
25	23, 24	bitr4di 288	. . . . 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 47453	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
31	29, 30	sylib 217	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
32	15, 28, 31	mpjaodan 957	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃*wmo 2532  Vcvv 3474  ∅c0 4321  {csn 4627   × cxp 5673  ⟶wf 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is referenced by:  functhinclem1  47614  functhinclem3  47616