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Theorem imasetpreimafvbijlemfv 44380
Description: Lemma for imasetpreimafvbij 44384: 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 6439 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
21anim1i 618 . . . 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 44378 . . 3 ((Fun 𝐹𝑌𝑃) → (𝐻𝑌) = (𝐹𝑌))
73, 6syl 17 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑌))
813ad2ant1 1134 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → Fun 𝐹)
9 funiunfv 7019 . . 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 488 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑦𝑌)
15 simpl3 1194 . . . . 5 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → 𝑋𝑌)
164elsetpreimafveqfv 44370 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑌𝑃𝑦𝑌𝑋𝑌)) → (𝐹𝑦) = (𝐹𝑋))
1712, 13, 14, 15, 16syl13anc 1373 . . . 4 (((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) ∧ 𝑦𝑌) → (𝐹𝑦) = (𝐹𝑋))
1817ralrimiva 3096 . . 3 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
19 fveq2 6675 . . . 4 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2019iuneqconst 4893 . . 3 ((𝑋𝑌 ∧ ∀𝑦𝑌 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
2111, 18, 20syl2anc 587 . 2 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → 𝑦𝑌 (𝐹𝑦) = (𝐹𝑋))
227, 10, 213eqtr2d 2779 1 ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  {cab 2716  wral 3053  wrex 3054  {csn 4517   cuni 4797   ciun 4882  cmpt 5111  ccnv 5525  cima 5529  Fun wfun 6334   Fn wfn 6335  cfv 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-fv 6348
This theorem is referenced by:  imasetpreimafvbijlemfv1  44381
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