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Theorem fundcmpsurinjlem3 45682
Description: Lemma 3 for fundcmpsurinj 45691. (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 6013 . . . 4 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
43unieqd 4883 . . 3 (𝑝 = 𝑋 (𝐹𝑝) = (𝐹𝑋))
54adantl 483 . 2 (((Fun 𝐹𝑋𝑃) ∧ 𝑝 = 𝑋) → (𝐹𝑝) = (𝐹𝑋))
6 simpr 486 . 2 ((Fun 𝐹𝑋𝑃) → 𝑋𝑃)
7 funimaexg 6591 . . 3 ((Fun 𝐹𝑋𝑃) → (𝐹𝑋) ∈ V)
87uniexd 7683 . 2 ((Fun 𝐹𝑋𝑃) → (𝐹𝑋) ∈ V)
92, 5, 6, 8fvmptd 6959 1 ((Fun 𝐹𝑋𝑃) → (𝐻𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3070  Vcvv 3447  {csn 4590   cuni 4869  cmpt 5192  ccnv 5636  cima 5640  Fun wfun 6494  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-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-ral 3062  df-rex 3071  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-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  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-fv 6508
This theorem is referenced by:  imasetpreimafvbijlemfv  45684
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