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

Theorem fliftval 7309
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 480 . 2 ((𝜑𝑌𝑋) → Fun 𝐹)
3 simpr 484 . . . 4 ((𝜑𝑌𝑋) → 𝑌𝑋)
4 eqidd 2727 . . . . 5 (𝜑𝐷 = 𝐷)
5 eqidd 2727 . . . . 5 (𝑌𝑋𝐶 = 𝐶)
64, 5anim12ci 613 . . . 4 ((𝜑𝑌𝑋) → (𝐶 = 𝐶𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (𝑥 = 𝑌𝐴 = 𝐶)
87eqeq2d 2737 . . . . . 6 (𝑥 = 𝑌 → (𝐶 = 𝐴𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (𝑥 = 𝑌𝐵 = 𝐷)
109eqeq2d 2737 . . . . . 6 (𝑥 = 𝑌 → (𝐷 = 𝐵𝐷 = 𝐷))
118, 10anbi12d 630 . . . . 5 (𝑥 = 𝑌 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
1211rspcev 3606 . . . 4 ((𝑌𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
133, 6, 12syl2anc 583 . . 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 7302 . . . 4 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1817adantr 480 . . 3 ((𝜑𝑌𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1913, 18mpbird 257 . 2 ((𝜑𝑌𝑋) → 𝐶𝐹𝐷)
20 funbrfv 6936 . 2 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
212, 19, 20sylc 65 1 ((𝜑𝑌𝑋) → (𝐹𝐶) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  cop 4629   class class class wbr 5141  cmpt 5224  ran crn 5670  Fun wfun 6531  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by:  qliftval  8802  cygznlem2  21463  pi1xfrval  24936  pi1coval  24942
  Copyright terms: Public domain W3C validator