Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fundcmpsurinjlem2 Structured version   Visualization version   GIF version

Theorem fundcmpsurinjlem2 47386
Description: Lemma 2 for fundcmpsurinj 47396. (Contributed by AV, 4-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.g 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
Assertion
Ref Expression
fundcmpsurinjlem2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑉
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝐺(𝑥,𝑧)   𝑉(𝑧)

Proof of Theorem fundcmpsurinjlem2
StepHypRef Expression
1 fnex 7237 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2 cnvexg 7946 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
3 imaexg 7935 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑥)}) ∈ V)
41, 2, 33syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑥)}) ∈ V)
54ralrimivw 3150 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V)
6 fundcmpsurinj.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
76fnmpt 6708 . . 3 (∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V → 𝐺 Fn 𝐴)
85, 7syl 17 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺 Fn 𝐴)
9 fundcmpsurinj.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
109, 6fundcmpsurinjlem1 47385 . 2 ran 𝐺 = 𝑃
11 df-fo 6567 . 2 (𝐺:𝐴onto𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
128, 10, 11sylanblrc 590 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  {csn 4626  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  ontowfo 6559  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fundcmpsurbijinjpreimafv  47394
  Copyright terms: Public domain W3C validator