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Theorem fliftval 6729
 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 472 . 2 ((𝜑𝑌𝑋) → Fun 𝐹)
3 simpr 479 . . . 4 ((𝜑𝑌𝑋) → 𝑌𝑋)
4 eqidd 2761 . . . . 5 (𝜑𝐷 = 𝐷)
5 eqidd 2761 . . . . 5 (𝑌𝑋𝐶 = 𝐶)
64, 5anim12ci 592 . . . 4 ((𝜑𝑌𝑋) → (𝐶 = 𝐶𝐷 = 𝐷))
7 fliftval.4 . . . . . . 7 (𝑥 = 𝑌𝐴 = 𝐶)
87eqeq2d 2770 . . . . . 6 (𝑥 = 𝑌 → (𝐶 = 𝐴𝐶 = 𝐶))
9 fliftval.5 . . . . . . 7 (𝑥 = 𝑌𝐵 = 𝐷)
109eqeq2d 2770 . . . . . 6 (𝑥 = 𝑌 → (𝐷 = 𝐵𝐷 = 𝐷))
118, 10anbi12d 749 . . . . 5 (𝑥 = 𝑌 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
1211rspcev 3449 . . . 4 ((𝑌𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
133, 6, 12syl2anc 696 . . 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 6722 . . . 4 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1817adantr 472 . . 3 ((𝜑𝑌𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1913, 18mpbird 247 . 2 ((𝜑𝑌𝑋) → 𝐶𝐹𝐷)
20 funbrfv 6395 . 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 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  ⟨cop 4327   class class class wbr 4804   ↦ cmpt 4881  ran crn 5267  Fun wfun 6043  ‘cfv 6049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057 This theorem is referenced by:  qliftval  8003  cygznlem2  20119  pi1xfrval  23054  pi1coval  23060
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