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Theorem imasetpreimafvbijlemfv 47327
Description: Lemma for imasetpreimafvbij 47331: 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 6669 . . . . 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 47325 . . 3 ((Fun 𝐹𝑌𝑃) → (𝐻𝑌) = (𝐹𝑌))
73, 6syl 17 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑌))
813ad2ant1 1132 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → Fun 𝐹)
9 funiunfv 7268 . . 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 484 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑦𝑌)
15 simpl3 1192 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑋𝑌)
164elsetpreimafveqfv 47317 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑌𝑃𝑦𝑌𝑋𝑌)) → (𝐹𝑦) = (𝐹𝑋))
1712, 13, 14, 15, 16syl13anc 1371 . . . 4 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → (𝐹𝑦) = (𝐹𝑋))
1817ralrimiva 3144 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
19 fveq2 6907 . . . 4 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2019iuneqconst 5008 . . 3 ((𝑋𝑌 ∧ ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
2111, 18, 20syl2anc 584 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
227, 10, 213eqtr2d 2781 1 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  {csn 4631   cuni 4912   ciun 4996  cmpt 5231  ccnv 5688  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  imasetpreimafvbijlemfv1  47328
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