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Theorem fliftfun 5557
Description: The function 𝐹 is the unique function defined by 𝐹𝐴 = 𝐵, provided that the well-definedness condition holds. (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)
fliftfun.4 (𝑥 = 𝑦𝐴 = 𝐶)
fliftfun.5 (𝑥 = 𝑦𝐵 = 𝐷)
Assertion
Ref Expression
fliftfun (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝑦,𝑅   𝑥,𝐷   𝑦,𝐹   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝐹(𝑥)

Proof of Theorem fliftfun
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . . 3 𝑥𝜑
2 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
3 nfmpt1 3923 . . . . . 6 𝑥(𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
43nfrn 4668 . . . . 5 𝑥ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
52, 4nfcxfr 2225 . . . 4 𝑥𝐹
65nffun 5024 . . 3 𝑥Fun 𝐹
7 fveq2 5289 . . . . . . 7 (𝐴 = 𝐶 → (𝐹𝐴) = (𝐹𝐶))
8 simplr 497 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → Fun 𝐹)
9 flift.2 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐴𝑅)
10 flift.3 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐵𝑆)
112, 9, 10fliftel1 5555 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
1211ad2ant2r 493 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝐹𝐵)
13 funbrfv 5327 . . . . . . . . 9 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
148, 12, 13sylc 61 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐴) = 𝐵)
15 simprr 499 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
16 eqidd 2089 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 = 𝐶)
17 eqidd 2089 . . . . . . . . . . 11 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 = 𝐷)
18 fliftfun.4 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 = 𝐶)
1918eqeq2d 2099 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐶 = 𝐴𝐶 = 𝐶))
20 fliftfun.5 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐵 = 𝐷)
2120eqeq2d 2099 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐷 = 𝐵𝐷 = 𝐷))
2219, 21anbi12d 457 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 = 𝐴𝐷 = 𝐵) ↔ (𝐶 = 𝐶𝐷 = 𝐷)))
2322rspcev 2722 . . . . . . . . . . 11 ((𝑦𝑋 ∧ (𝐶 = 𝐶𝐷 = 𝐷)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
2415, 16, 17, 23syl12anc 1172 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵))
252, 9, 10fliftel 5554 . . . . . . . . . . 11 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2625ad2antrr 472 . . . . . . . . . 10 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
2724, 26mpbird 165 . . . . . . . . 9 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶𝐹𝐷)
28 funbrfv 5327 . . . . . . . . 9 (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹𝐶) = 𝐷))
298, 27, 28sylc 61 . . . . . . . 8 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝐶) = 𝐷)
3014, 29eqeq12d 2102 . . . . . . 7 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐵 = 𝐷))
317, 30syl5ib 152 . . . . . 6 (((𝜑 ∧ Fun 𝐹) ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 = 𝐶𝐵 = 𝐷))
3231anassrs 392 . . . . 5 ((((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝐴 = 𝐶𝐵 = 𝐷))
3332ralrimiva 2446 . . . 4 (((𝜑 ∧ Fun 𝐹) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))
3433exp31 356 . . 3 (𝜑 → (Fun 𝐹 → (𝑥𝑋 → ∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷))))
351, 6, 34ralrimd 2451 . 2 (𝜑 → (Fun 𝐹 → ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
362, 9, 10fliftel 5554 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑢 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵)))
372, 9, 10fliftel 5554 . . . . . . . . . 10 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵)))
3818eqeq2d 2099 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑧 = 𝐴𝑧 = 𝐶))
3920eqeq2d 2099 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑣 = 𝐵𝑣 = 𝐷))
4038, 39anbi12d 457 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑣 = 𝐵) ↔ (𝑧 = 𝐶𝑣 = 𝐷)))
4140cbvrexv 2591 . . . . . . . . . 10 (∃𝑥𝑋 (𝑧 = 𝐴𝑣 = 𝐵) ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))
4237, 41syl6bb 194 . . . . . . . . 9 (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
4336, 42anbi12d 457 . . . . . . . 8 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
4443biimpd 142 . . . . . . 7 (𝜑 → ((𝑧𝐹𝑢𝑧𝐹𝑣) → (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷))))
45 reeanv 2536 . . . . . . . 8 (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) ↔ (∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)))
46 r19.29 2506 . . . . . . . . . 10 ((∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → ∃𝑥𝑋 (∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))))
47 r19.29 2506 . . . . . . . . . . . 12 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → ∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))))
48 eqtr2 2106 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝐴𝑧 = 𝐶) → 𝐴 = 𝐶)
4948ad2ant2r 493 . . . . . . . . . . . . . . . 16 (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐴 = 𝐶)
5049imim1i 59 . . . . . . . . . . . . . . 15 ((𝐴 = 𝐶𝐵 = 𝐷) → (((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝐵 = 𝐷))
5150imp 122 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝐵 = 𝐷)
52 simprlr 505 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝐵)
53 simprrr 507 . . . . . . . . . . . . . 14 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑣 = 𝐷)
5451, 52, 533eqtr4d 2130 . . . . . . . . . . . . 13 (((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5554rexlimivw 2485 . . . . . . . . . . . 12 (∃𝑦𝑋 ((𝐴 = 𝐶𝐵 = 𝐷) ∧ ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5647, 55syl 14 . . . . . . . . . . 11 ((∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5756rexlimivw 2485 . . . . . . . . . 10 (∃𝑥𝑋 (∀𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5846, 57syl 14 . . . . . . . . 9 ((∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷))) → 𝑢 = 𝑣)
5958ex 113 . . . . . . . 8 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → (∃𝑥𝑋𝑦𝑋 ((𝑧 = 𝐴𝑢 = 𝐵) ∧ (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6045, 59syl5bir 151 . . . . . . 7 (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((∃𝑥𝑋 (𝑧 = 𝐴𝑢 = 𝐵) ∧ ∃𝑦𝑋 (𝑧 = 𝐶𝑣 = 𝐷)) → 𝑢 = 𝑣))
6144, 60syl9 71 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6261alrimdv 1804 . . . . 5 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6362alrimdv 1804 . . . 4 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
6463alrimdv 1804 . . 3 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
652, 9, 10fliftrel 5553 . . . . 5 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
66 relxp 4535 . . . . 5 Rel (𝑅 × 𝑆)
67 relss 4513 . . . . 5 (𝐹 ⊆ (𝑅 × 𝑆) → (Rel (𝑅 × 𝑆) → Rel 𝐹))
6865, 66, 67mpisyl 1380 . . . 4 (𝜑 → Rel 𝐹)
69 dffun2 5012 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7069baib 866 . . . 4 (Rel 𝐹 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7168, 70syl 14 . . 3 (𝜑 → (Fun 𝐹 ↔ ∀𝑧𝑢𝑣((𝑧𝐹𝑢𝑧𝐹𝑣) → 𝑢 = 𝑣)))
7264, 71sylibrd 167 . 2 (𝜑 → (∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷) → Fun 𝐹))
7335, 72impbid 127 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wcel 1438  wral 2359  wrex 2360  wss 2997  cop 3444   class class class wbr 3837  cmpt 3891   × cxp 4426  ran crn 4429  Rel wrel 4433  Fun wfun 4996  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010
This theorem is referenced by:  fliftfund  5558  fliftfuns  5559  qliftfun  6354
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