Theorem mofeu 49089

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 6716	. . . . 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 6642	. . . 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 5645	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10		xp0 5724	. . . . . . 7 ⊢ (𝐴 × ∅) = ∅
11	9, 10	eqtrdi 2787	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
12	8, 11	eqtrid 2783	. . . . 5 ⊢ (𝐵 = ∅ → 𝐺 = ∅)
13	12	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐺 = ∅)
14	13	eqeq2d 2747	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹 = 𝐺 ↔ 𝐹 = ∅))
15	5, 7, 14	3bitr4d 311	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
16		19.42v 1954	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17		fconst2g 7149	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
18	17	elv 3445	. . . . . . 7 ⊢ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19		feq3 6642	. . . . . . . 8 ⊢ (𝐵 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶{𝑦}))
20		xpeq2 5645	. . . . . . . . 9 ⊢ (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
21	20	eqeq2d 2747	. . . . . . . 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 2749	. . . . . 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 1931	. . 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 49057	. . 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 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2537  Vcvv 3440  ∅c0 4285  {csn 4580   × cxp 5622  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  functhinclem1  49685  functhinclem3  49687