Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftfuns Structured version   Visualization version   GIF version

Theorem fliftfuns 6464
 Description: The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2 ((𝜑𝑥𝑋) → 𝐴𝑅)
flift.3 ((𝜑𝑥𝑋) → 𝐵𝑆)
Assertion
Ref Expression
fliftfuns (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝑧,𝑦,𝑅   𝑦,𝐹,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftfuns
StepHypRef Expression
1 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
2 nfcv 2751 . . . . 5 𝑦𝐴, 𝐵
3 nfcsb1v 3515 . . . . . 6 𝑥𝑦 / 𝑥𝐴
4 nfcsb1v 3515 . . . . . 6 𝑥𝑦 / 𝑥𝐵
53, 4nfop 4356 . . . . 5 𝑥𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵
6 csbeq1a 3508 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
7 csbeq1a 3508 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
86, 7opeq12d 4348 . . . . 5 (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
92, 5, 8cbvmpt 4677 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
109rneqi 5273 . . 3 ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
111, 10eqtri 2632 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨𝑦 / 𝑥𝐴, 𝑦 / 𝑥𝐵⟩)
12 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
1312ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑅)
143nfel1 2765 . . . 4 𝑥𝑦 / 𝑥𝐴𝑅
156eleq1d 2672 . . . 4 (𝑥 = 𝑦 → (𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1614, 15rspc 3276 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑅𝑦 / 𝑥𝐴𝑅))
1713, 16mpan9 485 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑅)
18 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
1918ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝑋 𝐵𝑆)
204nfel1 2765 . . . 4 𝑥𝑦 / 𝑥𝐵𝑆
217eleq1d 2672 . . . 4 (𝑥 = 𝑦 → (𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2220, 21rspc 3276 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐵𝑆𝑦 / 𝑥𝐵𝑆))
2319, 22mpan9 485 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐵𝑆)
24 csbeq1 3502 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
25 csbeq1 3502 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
2611, 17, 23, 24, 25fliftfun 6462 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator