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Theorem imasetpreimafvbijlemfv 47389
Description: Lemma for imasetpreimafvbij 47393: 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 6668 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
21anim1i 615 . . . 4 ((𝐹 Fn 𝐴𝑌𝑃) → (Fun 𝐹𝑌𝑃))
323adant3 1133 . . 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 47387 . . 3 ((Fun 𝐹𝑌𝑃) → (𝐻𝑌) = (𝐹𝑌))
73, 6syl 17 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑌))
813ad2ant1 1134 . . 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 1139 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑋𝑌)
12 simpl1 1192 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝐹 Fn 𝐴)
13 simpl2 1193 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑌𝑃)
14 simpr 484 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑦𝑌)
15 simpl3 1194 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑋𝑌)
164elsetpreimafveqfv 47379 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑌𝑃𝑦𝑌𝑋𝑌)) → (𝐹𝑦) = (𝐹𝑋))
1712, 13, 14, 15, 16syl13anc 1374 . . . 4 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → (𝐹𝑦) = (𝐹𝑋))
1817ralrimiva 3146 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
19 fveq2 6906 . . . 4 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2019iuneqconst 5003 . . 3 ((𝑋𝑌 ∧ ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
2111, 18, 20syl2anc 584 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
227, 10, 213eqtr2d 2783 1 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {csn 4626   cuni 4907   ciun 4991  cmpt 5225  ccnv 5684  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  imasetpreimafvbijlemfv1  47390
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