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Theorem fundcmpsurinjlem3 43932
 Description: Lemma 3 for fundcmpsurinj 43941. (Contributed by AV, 3-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.h 𝐻 = (𝑝𝑃 (𝐹𝑝))
Assertion
Ref Expression
fundcmpsurinjlem3 ((Fun 𝐹𝑋𝑃) → (𝐻𝑋) = (𝐹𝑋))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝐹,𝑝   𝑃,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐴(𝑝)   𝑃(𝑥,𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑋(𝑥,𝑧)

Proof of Theorem fundcmpsurinjlem3
StepHypRef Expression
1 fundcmpsurinj.h . . 3 𝐻 = (𝑝𝑃 (𝐹𝑝))
21a1i 11 . 2 ((Fun 𝐹𝑋𝑃) → 𝐻 = (𝑝𝑃 (𝐹𝑝)))
3 imaeq2 5892 . . . 4 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
43unieqd 4814 . . 3 (𝑝 = 𝑋 (𝐹𝑝) = (𝐹𝑋))
54adantl 485 . 2 (((Fun 𝐹𝑋𝑃) ∧ 𝑝 = 𝑋) → (𝐹𝑝) = (𝐹𝑋))
6 simpr 488 . 2 ((Fun 𝐹𝑋𝑃) → 𝑋𝑃)
7 funimaexg 6410 . . 3 ((Fun 𝐹𝑋𝑃) → (𝐹𝑋) ∈ V)
87uniexd 7450 . 2 ((Fun 𝐹𝑋𝑃) → (𝐹𝑋) ∈ V)
92, 5, 6, 8fvmptd 6752 1 ((Fun 𝐹𝑋𝑃) → (𝐻𝑋) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441  {csn 4525  ∪ cuni 4800   ↦ cmpt 5110  ◡ccnv 5518   “ cima 5522  Fun wfun 6318  ‘cfv 6324 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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332 This theorem is referenced by:  imasetpreimafvbijlemfv  43934
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