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Theorem fliftf 7312
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 486 . . . . 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 7306 . . . . . . . . . 10 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
65exbidv 1925 . . . . . . . . 9 (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
76adantr 482 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
8 rexcom4 3286 . . . . . . . . 9 (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵))
9 19.42v 1958 . . . . . . . . . . . 12 (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
10 elisset 2816 . . . . . . . . . . . . . 14 (𝐵𝑆 → ∃𝑧 𝑧 = 𝐵)
114, 10syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∃𝑧 𝑧 = 𝐵)
1211biantrud 533 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
139, 12bitr4id 290 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
1413rexbidva 3177 . . . . . . . . . 10 (𝜑 → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1514adantr 482 . . . . . . . . 9 ((𝜑 ∧ Fun 𝐹) → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
168, 15bitr3id 285 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
177, 16bitrd 279 . . . . . . 7 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1817abbidv 2802 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴})
19 df-dm 5687 . . . . . 6 dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20 eqid 2733 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120rnmpt 5955 . . . . . 6 ran (𝑥𝑋𝐴) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴}
2218, 19, 213eqtr4g 2798 . . . . 5 ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥𝑋𝐴))
23 df-fn 6547 . . . . 5 (𝐹 Fn ran (𝑥𝑋𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥𝑋𝐴)))
241, 22, 23sylanbrc 584 . . . 4 ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥𝑋𝐴))
252, 3, 4fliftrel 7305 . . . . . . 7 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 482 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 5939 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 17 . . . . 5 ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 6172 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29sstrdi 3995 . . . 4 ((𝜑 ∧ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 6548 . . . 4 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥𝑋𝐴) ∧ ran 𝐹𝑆))
3224, 30, 31sylanbrc 584 . . 3 ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥𝑋𝐴)⟶𝑆)
3332ex 414 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
34 ffun 6721 . 2 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 → Fun 𝐹)
3533, 34impbid1 224 1 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  wss 3949  cop 4635   class class class wbr 5149  cmpt 5232   × cxp 5675  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  qliftf  8799  cygznlem2a  21123  pi1xfrf  24569  pi1cof  24575
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