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Theorem fliftf 7260
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 485 . . . . 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 7254 . . . . . . . . . 10 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
65exbidv 1924 . . . . . . . . 9 (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
76adantr 481 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵)))
8 rexcom4 3271 . . . . . . . . 9 (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵))
9 19.42v 1957 . . . . . . . . . . . 12 (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
10 elisset 2819 . . . . . . . . . . . . . 14 (𝐵𝑆 → ∃𝑧 𝑧 = 𝐵)
114, 10syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ∃𝑧 𝑧 = 𝐵)
1211biantrud 532 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
139, 12bitr4id 289 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∃𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
1413rexbidva 3173 . . . . . . . . . 10 (𝜑 → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1514adantr 481 . . . . . . . . 9 ((𝜑 ∧ Fun 𝐹) → (∃𝑥𝑋𝑧(𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
168, 15bitr3id 284 . . . . . . . 8 ((𝜑 ∧ Fun 𝐹) → (∃𝑧𝑥𝑋 (𝑦 = 𝐴𝑧 = 𝐵) ↔ ∃𝑥𝑋 𝑦 = 𝐴))
177, 16bitrd 278 . . . . . . 7 ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥𝑋 𝑦 = 𝐴))
1817abbidv 2805 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴})
19 df-dm 5643 . . . . . 6 dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20 eqid 2736 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120rnmpt 5910 . . . . . 6 ran (𝑥𝑋𝐴) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = 𝐴}
2218, 19, 213eqtr4g 2801 . . . . 5 ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥𝑋𝐴))
23 df-fn 6499 . . . . 5 (𝐹 Fn ran (𝑥𝑋𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥𝑋𝐴)))
241, 22, 23sylanbrc 583 . . . 4 ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥𝑋𝐴))
252, 3, 4fliftrel 7253 . . . . . . 7 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
2625adantr 481 . . . . . 6 ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27 rnss 5894 . . . . . 6 (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
2826, 27syl 17 . . . . 5 ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29 rnxpss 6124 . . . . 5 ran (𝑅 × 𝑆) ⊆ 𝑆
3028, 29sstrdi 3956 . . . 4 ((𝜑 ∧ Fun 𝐹) → ran 𝐹𝑆)
31 df-f 6500 . . . 4 (𝐹:ran (𝑥𝑋𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥𝑋𝐴) ∧ ran 𝐹𝑆))
3224, 30, 31sylanbrc 583 . . 3 ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥𝑋𝐴)⟶𝑆)
3332ex 413 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))
34 ffun 6671 . 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 396   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wrex 3073  wss 3910  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  Fun wfun 6490   Fn wfn 6491  wf 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-fun 6498  df-fn 6499  df-f 6500
This theorem is referenced by:  qliftf  8744  cygznlem2a  20974  pi1xfrf  24416  pi1cof  24422
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