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Theorem fliftval 5440
 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 261 . 2 ((𝜑𝑌𝑋) → Fun 𝐹)
3 simpr 103 . . . 4 ((𝜑𝑌𝑋) → 𝑌𝑋)
4 eqidd 2041 . . . . 5 (𝜑𝐷 = 𝐷)
5 eqidd 2041 . . . . 5 (𝑌𝑋𝐶 = 𝐶)
64, 5anim12ci 322 . . . 4 ((𝜑𝑌𝑋) → (𝐶 = 𝐶𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (𝑥 = 𝑌𝐴 = 𝐶)
87eqeq2d 2051 . . . . . 6 (𝑥 = 𝑌 → (𝐶 = 𝐴𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (𝑥 = 𝑌𝐵 = 𝐷)
109eqeq2d 2051 . . . . . 6 (𝑥 = 𝑌 → (𝐷 = 𝐵𝐷 = 𝐷))
118, 10anbi12d 442 . . . . 5 (𝑥 = 𝑌 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
1211rspcev 2656 . . . 4 ((𝑌𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
133, 6, 12syl2anc 391 . . 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 5433 . . . 4 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1817adantr 261 . . 3 ((𝜑𝑌𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1913, 18mpbird 156 . 2 ((𝜑𝑌𝑋) → 𝐶𝐹𝐷)
20 funbrfv 5212 . 2 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
212, 19, 20sylc 56 1 ((𝜑𝑌𝑋) → (𝐹𝐶) = 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764   ↦ cmpt 3818  ran crn 4346  Fun wfun 4896  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910 This theorem is referenced by:  qliftval  6192
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