Theorem mofeu 48836

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 406	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 = ∅)
3		f00 6742	. . . . 5 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	rbaib 538	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
5	2, 4	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
6		feq3 6668	. . . 4 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
7	6	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
8		mofeu.1	. . . . . 6 ⊢ 𝐺 = (𝐴 × 𝐵)
9		xpeq2 5659	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10		xp0 6131	. . . . . . 7 ⊢ (𝐴 × ∅) = ∅
11	9, 10	eqtrdi 2780	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
12	8, 11	eqtrid 2776	. . . . 5 ⊢ (𝐵 = ∅ → 𝐺 = ∅)
13	12	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐺 = ∅)
14	13	eqeq2d 2740	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹 = 𝐺 ↔ 𝐹 = ∅))
15	5, 7, 14	3bitr4d 311	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
16		19.42v 1953	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17		fconst2g 7177	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
18	17	elv 3452	. . . . . . 7 ⊢ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19		feq3 6668	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶{𝑦}))
20		xpeq2 5659	. . . . . . . . 9 ⊢ (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
21	20	eqeq2d 2740	. . . . . . . 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 258	. . . . . 6 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = (𝐴 × 𝐵)))
24	8	eqeq2i 2742	. . . . . 6 ⊢ (𝐹 = 𝐺 ↔ 𝐹 = (𝐴 × 𝐵))
25	23, 24	bitr4di 289	. . . . 5 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
26	25	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
27	26	exlimiv 1930	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
28	16, 27	sylbir 235	. 2 ⊢ ((𝜑 ∧ ∃𝑦 𝐵 = {𝑦}) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
29		mofeu.3	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
30		mo0sn 48804	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
31	29, 30	sylib 218	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
32	15, 28, 31	mpjaodan 960	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  Vcvv 3447  ∅c0 4296  {csn 4589   × cxp 5636  ⟶wf 6507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  functhinclem1  49433  functhinclem3  49435