ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fliftval GIF version

Theorem fliftval 5669
Description: The value of the function 𝐹. (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)
fliftval.4 (𝑥 = 𝑌𝐴 = 𝐶)
fliftval.5 (𝑥 = 𝑌𝐵 = 𝐷)
fliftval.6 (𝜑 → Fun 𝐹)
Assertion
Ref Expression
fliftval ((𝜑𝑌𝑋) → (𝐹𝐶) = 𝐷)
Distinct variable groups:   𝑥,𝐶   𝑥,𝑅   𝑥,𝑌   𝑥,𝐷   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftval
StepHypRef Expression
1 fliftval.6 . . 3 (𝜑 → Fun 𝐹)
21adantr 274 . 2 ((𝜑𝑌𝑋) → Fun 𝐹)
3 simpr 109 . . . 4 ((𝜑𝑌𝑋) → 𝑌𝑋)
4 eqidd 2118 . . . . 5 (𝜑𝐷 = 𝐷)
5 eqidd 2118 . . . . 5 (𝑌𝑋𝐶 = 𝐶)
64, 5anim12ci 337 . . . 4 ((𝜑𝑌𝑋) → (𝐶 = 𝐶𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (𝑥 = 𝑌𝐴 = 𝐶)
87eqeq2d 2129 . . . . . 6 (𝑥 = 𝑌 → (𝐶 = 𝐴𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (𝑥 = 𝑌𝐵 = 𝐷)
109eqeq2d 2129 . . . . . 6 (𝑥 = 𝑌 → (𝐷 = 𝐵𝐷 = 𝐷))
118, 10anbi12d 464 . . . . 5 (𝑥 = 𝑌 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
1211rspcev 2763 . . . 4 ((𝑌𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
133, 6, 12syl2anc 408 . . 3 ((𝜑𝑌𝑋) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
14 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
15 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
16 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
1714, 15, 16fliftel 5662 . . . 4 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1817adantr 274 . . 3 ((𝜑𝑌𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1913, 18mpbird 166 . 2 ((𝜑𝑌𝑋) → 𝐶𝐹𝐷)
20 funbrfv 5428 . 2 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
212, 19, 20sylc 62 1 ((𝜑𝑌𝑋) → (𝐹𝐶) = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wrex 2394  cop 3500   class class class wbr 3899  cmpt 3959  ran crn 4510  Fun wfun 5087  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by:  qliftval  6483
  Copyright terms: Public domain W3C validator