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Theorem fundcmpsurinjlem2 46552
Description: Lemma 2 for fundcmpsurinj 46562. (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 7210 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2 cnvexg 7908 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
3 imaexg 7899 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑥)}) ∈ V)
41, 2, 33syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑥)}) ∈ V)
54ralrimivw 3142 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V)
6 fundcmpsurinj.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
76fnmpt 6680 . . 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 46551 . 2 ran 𝐺 = 𝑃
11 df-fo 6539 . 2 (𝐺:𝐴onto𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
128, 10, 11sylanblrc 589 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2701  wral 3053  wrex 3062  Vcvv 3466  {csn 4620  cmpt 5221  ccnv 5665  ran crn 5667  cima 5669   Fn wfn 6528  ontowfo 6531  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541
This theorem is referenced by:  fundcmpsurbijinjpreimafv  46560
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