Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mofeu Structured version   Visualization version   GIF version

Theorem mofeu 46904
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 𝑥𝐵)
Assertion
Ref Expression
mofeu (𝜑 → (𝐹:𝐴𝐵𝐹 = 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.
StepHypRef Expression
1 mofeu.2 . . . . 5 (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
21imp 407 . . . 4 ((𝜑𝐵 = ∅) → 𝐴 = ∅)
3 f00 6724 . . . . 5 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
43rbaib 539 . . . 4 (𝐴 = ∅ → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
52, 4syl 17 . . 3 ((𝜑𝐵 = ∅) → (𝐹:𝐴⟶∅ ↔ 𝐹 = ∅))
6 feq3 6651 . . . 4 (𝐵 = ∅ → (𝐹:𝐴𝐵𝐹:𝐴⟶∅))
76adantl 482 . . 3 ((𝜑𝐵 = ∅) → (𝐹:𝐴𝐵𝐹:𝐴⟶∅))
8 mofeu.1 . . . . . 6 𝐺 = (𝐴 × 𝐵)
9 xpeq2 5654 . . . . . . 7 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
10 xp0 6110 . . . . . . 7 (𝐴 × ∅) = ∅
119, 10eqtrdi 2792 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
128, 11eqtrid 2788 . . . . 5 (𝐵 = ∅ → 𝐺 = ∅)
1312adantl 482 . . . 4 ((𝜑𝐵 = ∅) → 𝐺 = ∅)
1413eqeq2d 2747 . . 3 ((𝜑𝐵 = ∅) → (𝐹 = 𝐺𝐹 = ∅))
155, 7, 143bitr4d 310 . 2 ((𝜑𝐵 = ∅) → (𝐹:𝐴𝐵𝐹 = 𝐺))
16 19.42v 1957 . . 3 (∃𝑦(𝜑𝐵 = {𝑦}) ↔ (𝜑 ∧ ∃𝑦 𝐵 = {𝑦}))
17 fconst2g 7152 . . . . . . . 8 (𝑦 ∈ V → (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦})))
1817elv 3451 . . . . . . 7 (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))
19 feq3 6651 . . . . . . . 8 (𝐵 = {𝑦} → (𝐹:𝐴𝐵𝐹:𝐴⟶{𝑦}))
20 xpeq2 5654 . . . . . . . . 9 (𝐵 = {𝑦} → (𝐴 × 𝐵) = (𝐴 × {𝑦}))
2120eqeq2d 2747 . . . . . . . 8 (𝐵 = {𝑦} → (𝐹 = (𝐴 × 𝐵) ↔ 𝐹 = (𝐴 × {𝑦})))
2219, 21bibi12d 345 . . . . . . 7 (𝐵 = {𝑦} → ((𝐹:𝐴𝐵𝐹 = (𝐴 × 𝐵)) ↔ (𝐹:𝐴⟶{𝑦} ↔ 𝐹 = (𝐴 × {𝑦}))))
2318, 22mpbiri 257 . . . . . 6 (𝐵 = {𝑦} → (𝐹:𝐴𝐵𝐹 = (𝐴 × 𝐵)))
248eqeq2i 2749 . . . . . 6 (𝐹 = 𝐺𝐹 = (𝐴 × 𝐵))
2523, 24bitr4di 288 . . . . 5 (𝐵 = {𝑦} → (𝐹:𝐴𝐵𝐹 = 𝐺))
2625adantl 482 . . . 4 ((𝜑𝐵 = {𝑦}) → (𝐹:𝐴𝐵𝐹 = 𝐺))
2726exlimiv 1933 . . 3 (∃𝑦(𝜑𝐵 = {𝑦}) → (𝐹:𝐴𝐵𝐹 = 𝐺))
2816, 27sylbir 234 . 2 ((𝜑 ∧ ∃𝑦 𝐵 = {𝑦}) → (𝐹:𝐴𝐵𝐹 = 𝐺))
29 mofeu.3 . . 3 (𝜑 → ∃*𝑥 𝑥𝐵)
30 mo0sn 46890 . . 3 (∃*𝑥 𝑥𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
3129, 30sylib 217 . 2 (𝜑 → (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
3215, 28, 31mpjaodan 957 1 (𝜑 → (𝐹:𝐴𝐵𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  ∃*wmo 2536  Vcvv 3445  c0 4282  {csn 4586   × cxp 5631  wf 6492
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504
This theorem is referenced by:  functhinclem1  47051  functhinclem3  47053
  Copyright terms: Public domain W3C validator