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Theorem fundcmpsurinjlem2 45681
Description: Lemma 2 for fundcmpsurinj 45691. (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 7171 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2 cnvexg 7865 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
3 imaexg 7856 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑥)}) ∈ V)
41, 2, 33syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑥)}) ∈ V)
54ralrimivw 3144 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V)
6 fundcmpsurinj.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
76fnmpt 6645 . . 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 45680 . 2 ran 𝐺 = 𝑃
11 df-fo 6506 . 2 (𝐺:𝐴onto𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
128, 10, 11sylanblrc 591 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3447  {csn 4590  cmpt 5192  ccnv 5636  ran crn 5638  cima 5640   Fn wfn 6495  ontowfo 6498  cfv 6500
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508
This theorem is referenced by:  fundcmpsurbijinjpreimafv  45689
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