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Theorem fliftf 5560
Description: The domain and range 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)
Assertion
Ref Expression
fliftf (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftf
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . 5 ((𝜑 ∧ Fun 𝐹) → Fun 𝐹)
2 flift.1 . . . . . . . . . . 11 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
3 flift.2 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐴𝑅)
4 flift.3 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝐵𝑆)
52, 3, 4fliftel 5554 . . . . . . . . . 10 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
65exbidv 1753 . . . . . . . . 9 (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
76adantr 270 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
8 rexcom4 2642 . . . . . . . . 9 (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵))
9 elisset 2633 . . . . . . . . . . . . . 14 (𝐵𝑆 → ∃𝑧 𝑧 = 𝐵)
104, 9syl 14 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∃𝑧 𝑧 = 𝐵)
1110biantrud 298 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
12 19.42v 1834 . . . . . . . . . . . 12 (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
1311, 12syl6rbbr 197 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
1413rexbidva 2377 . . . . . . . . . 10 (𝜑 → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1514adantr 270 . . . . . . . . 9 ((𝜑 ∧ Fun 𝐹) → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
168, 15syl5bbr 192 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
177, 16bitrd 186 . . . . . . 7 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1817abbidv 2205 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴})
19 df-dm 4438 . . . . . 6 dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20 eqid 2088 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120rnmpt 4671 . . . . . 6 ran (𝑥𝑋𝐴) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴}
2218, 19, 213eqtr4g 2145 . . . . 5 ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥𝑋𝐴))
23 df-fn 5005 . . . . 5 (𝐹 Fn ran (𝑥𝑋𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥𝑋𝐴)))
241, 22, 23sylanbrc 408 . . . 4 ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥𝑋𝐴))
252, 3, 4fliftrel 5553 . . . . . . 7 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 270 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 4653 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 14 . . . . 5 ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 4849 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29syl6ss 3035 . . . 4 ((𝜑 ∧ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 5006 . . . 4 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥𝑋𝐴) ∧ ran 𝐹𝑆))
3224, 30, 31sylanbrc 408 . . 3 ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥𝑋𝐴)⟶𝑆)
3332ex 113 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
34 ffun 5150 . 2 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 → Fun 𝐹)
3533, 34impbid1 140 1 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wrex 2360  wss 2997  cop 3444   class class class wbr 3837  cmpt 3891   × cxp 4426  dom cdm 4428  ran crn 4429  Fun wfun 4996   Fn wfn 4997  wf 4998
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-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:  qliftf  6357
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