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Theorem fundcmpsurinjlem2 47324
Description: Lemma 2 for fundcmpsurinj 47334. (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 7947 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
3 imaexg 7936 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑥)}) ∈ V)
41, 2, 33syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑥)}) ∈ V)
54ralrimivw 3148 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V)
6 fundcmpsurinj.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
76fnmpt 6709 . . 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 47323 . 2 ran 𝐺 = 𝑃
11 df-fo 6569 . 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 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  {csn 4631  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  fundcmpsurbijinjpreimafv  47332
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