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Theorem imasetpreimafvbijlemfv1 46666
Description: Lemma for imasetpreimafvbij 46669: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (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
imasetpreimafvbijlemfv1 ((𝐹 Fn 𝐴𝑋𝑃) → ∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝑋,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑦,𝐴   𝑦,𝐹   𝑦,𝑃   𝑥,𝑋,𝑦   𝑧,𝑋   𝑦,𝑝
Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑦,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemfv1
StepHypRef Expression
1 fundcmpsurinj.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
210nelsetpreimafv 46653 . . . 4 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3 elnelne2 3053 . . . . 5 ((𝑋𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅)
43expcom 413 . . . 4 (∅ ∉ 𝑃 → (𝑋𝑃𝑋 ≠ ∅))
52, 4syl 17 . . 3 (𝐹 Fn 𝐴 → (𝑋𝑃𝑋 ≠ ∅))
65imp 406 . 2 ((𝐹 Fn 𝐴𝑋𝑃) → 𝑋 ≠ ∅)
7 simpr 484 . . . . . 6 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → 𝑦𝑋)
8 fundcmpsurinj.h . . . . . . . 8 𝐻 = (𝑝𝑃 (𝐹𝑝))
91, 8imasetpreimafvbijlemfv 46665 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝑃𝑦𝑋) → (𝐻𝑋) = (𝐹𝑦))
1093expa 1116 . . . . . 6 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → (𝐻𝑋) = (𝐹𝑦))
117, 10jca 511 . . . . 5 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → (𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦)))
1211ex 412 . . . 4 ((𝐹 Fn 𝐴𝑋𝑃) → (𝑦𝑋 → (𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦))))
1312eximdv 1913 . . 3 ((𝐹 Fn 𝐴𝑋𝑃) → (∃𝑦 𝑦𝑋 → ∃𝑦(𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦))))
14 n0 4342 . . 3 (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦𝑋)
15 df-rex 3066 . . 3 (∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦) ↔ ∃𝑦(𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦)))
1613, 14, 153imtr4g 296 . 2 ((𝐹 Fn 𝐴𝑋𝑃) → (𝑋 ≠ ∅ → ∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦)))
176, 16mpd 15 1 ((𝐹 Fn 𝐴𝑋𝑃) → ∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wne 2935  wnel 3041  wrex 3065  c0 4318  {csn 4624   cuni 4903  cmpt 5225  ccnv 5671  cima 5675   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  imasetpreimafvbijlemf1  46667
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