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Theorem fundcmpsurinjlem2 44333
 Description: Lemma 2 for fundcmpsurinj 44343. (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 6977 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2 cnvexg 7640 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
3 imaexg 7631 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑥)}) ∈ V)
41, 2, 33syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑥)}) ∈ V)
54ralrimivw 3114 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ∀𝑥𝐴 (𝐹 “ {(𝐹𝑥)}) ∈ V)
6 fundcmpsurinj.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
76fnmpt 6476 . . 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 44332 . 2 ran 𝐺 = 𝑃
11 df-fo 6346 . 2 (𝐺:𝐴onto𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
128, 10, 11sylanblrc 593 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  Vcvv 3409  {csn 4525   ↦ cmpt 5116  ◡ccnv 5527  ran crn 5529   “ cima 5531   Fn wfn 6335  –onto→wfo 6338  ‘cfv 6340 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348 This theorem is referenced by:  fundcmpsurbijinjpreimafv  44341
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