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Theorem fliftval 7058
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 481 . 2 ((𝜑𝑌𝑋) → Fun 𝐹)
3 simpr 485 . . . 4 ((𝜑𝑌𝑋) → 𝑌𝑋)
4 eqidd 2819 . . . . 5 (𝜑𝐷 = 𝐷)
5 eqidd 2819 . . . . 5 (𝑌𝑋𝐶 = 𝐶)
64, 5anim12ci 613 . . . 4 ((𝜑𝑌𝑋) → (𝐶 = 𝐶𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (𝑥 = 𝑌𝐴 = 𝐶)
87eqeq2d 2829 . . . . . 6 (𝑥 = 𝑌 → (𝐶 = 𝐴𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (𝑥 = 𝑌𝐵 = 𝐷)
109eqeq2d 2829 . . . . . 6 (𝑥 = 𝑌 → (𝐷 = 𝐵𝐷 = 𝐷))
118, 10anbi12d 630 . . . . 5 (𝑥 = 𝑌 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
1211rspcev 3620 . . . 4 ((𝑌𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
133, 6, 12syl2anc 584 . . 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 7051 . . . 4 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1817adantr 481 . . 3 ((𝜑𝑌𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1913, 18mpbird 258 . 2 ((𝜑𝑌𝑋) → 𝐶𝐹𝐷)
20 funbrfv 6709 . 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 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  cop 4563   class class class wbr 5057  cmpt 5137  ran crn 5549  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  qliftval  8375  cygznlem2  20643  pi1xfrval  23585  pi1coval  23591
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