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Theorem imasetpreimafvbijlemfv 44854
Description: Lemma for imasetpreimafvbij 44858: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.h 𝐻 = (𝑝𝑃 (𝐹𝑝))
Assertion
Ref Expression
imasetpreimafvbijlemfv ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝑋,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑥,𝑋   𝑌,𝑝,𝑥,𝑧
Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑋(𝑧)

Proof of Theorem imasetpreimafvbijlemfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnfun 6533 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
21anim1i 615 . . . 4 ((𝐹 Fn 𝐴𝑌𝑃) → (Fun 𝐹𝑌𝑃))
323adant3 1131 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (Fun 𝐹𝑌𝑃))
4 fundcmpsurinj.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
5 fundcmpsurinj.h . . . 4 𝐻 = (𝑝𝑃 (𝐹𝑝))
64, 5fundcmpsurinjlem3 44852 . . 3 ((Fun 𝐹𝑌𝑃) → (𝐻𝑌) = (𝐹𝑌))
73, 6syl 17 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑌))
813ad2ant1 1132 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → Fun 𝐹)
9 funiunfv 7121 . . 3 (Fun 𝐹 𝑦𝑌 (𝐹𝑦) = (𝐹𝑌))
108, 9syl 17 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑌))
11 simp3 1137 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑋𝑌)
12 simpl1 1190 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝐹 Fn 𝐴)
13 simpl2 1191 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑌𝑃)
14 simpr 485 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑦𝑌)
15 simpl3 1192 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑋𝑌)
164elsetpreimafveqfv 44844 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑌𝑃𝑦𝑌𝑋𝑌)) → (𝐹𝑦) = (𝐹𝑋))
1712, 13, 14, 15, 16syl13anc 1371 . . . 4 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → (𝐹𝑦) = (𝐹𝑋))
1817ralrimiva 3103 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
19 fveq2 6774 . . . 4 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2019iuneqconst 4935 . . 3 ((𝑋𝑌 ∧ ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
2111, 18, 20syl2anc 584 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
227, 10, 213eqtr2d 2784 1 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  {csn 4561   cuni 4839   ciun 4924  cmpt 5157  ccnv 5588  cima 5592  Fun wfun 6427   Fn wfn 6428  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  imasetpreimafvbijlemfv1  44855
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