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Theorem fliftval 5383
Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
flift.2 ((φ x 𝑋) → A 𝑅)
flift.3 ((φ x 𝑋) → B 𝑆)
fliftval.4 (x = 𝑌A = 𝐶)
fliftval.5 (x = 𝑌B = 𝐷)
fliftval.6 (φ → Fun 𝐹)
Assertion
Ref Expression
fliftval ((φ 𝑌 𝑋) → (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 261 . 2 ((φ 𝑌 𝑋) → Fun 𝐹)
3 simpr 103 . . . 4 ((φ 𝑌 𝑋) → 𝑌 𝑋)
4 eqidd 2038 . . . . 5 (φ𝐷 = 𝐷)
5 eqidd 2038 . . . . 5 (𝑌 𝑋𝐶 = 𝐶)
64, 5anim12ci 322 . . . 4 ((φ 𝑌 𝑋) → (𝐶 = 𝐶 𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (x = 𝑌A = 𝐶)
87eqeq2d 2048 . . . . . 6 (x = 𝑌 → (𝐶 = A𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (x = 𝑌B = 𝐷)
109eqeq2d 2048 . . . . . 6 (x = 𝑌 → (𝐷 = B𝐷 = 𝐷))
118, 10anbi12d 442 . . . . 5 (x = 𝑌 → ((𝐶 = A 𝐷 = B) ↔ (𝐶 = 𝐶 𝐷 = 𝐷)))
1211rspcev 2650 . . . 4 ((𝑌 𝑋 (𝐶 = 𝐶 𝐷 = 𝐷)) → x 𝑋 (𝐶 = A 𝐷 = B))
133, 6, 12syl2anc 391 . . 3 ((φ 𝑌 𝑋) → x 𝑋 (𝐶 = A 𝐷 = B))
14 flift.1 . . . . 5 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
15 flift.2 . . . . 5 ((φ x 𝑋) → A 𝑅)
16 flift.3 . . . . 5 ((φ x 𝑋) → B 𝑆)
1714, 15, 16fliftel 5376 . . . 4 (φ → (𝐶𝐹𝐷x 𝑋 (𝐶 = A 𝐷 = B)))
1817adantr 261 . . 3 ((φ 𝑌 𝑋) → (𝐶𝐹𝐷x 𝑋 (𝐶 = A 𝐷 = B)))
1913, 18mpbird 156 . 2 ((φ 𝑌 𝑋) → 𝐶𝐹𝐷)
20 funbrfv 5155 . 2 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
212, 19, 20sylc 56 1 ((φ 𝑌 𝑋) → (𝐹𝐶) = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301  cop 3370   class class class wbr 3755  cmpt 3809  ran crn 4289  Fun wfun 4839  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  qliftval  6128
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