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Theorem imasetpreimafvbijlemfv1 45685
Description: Lemma for imasetpreimafvbij 45688: 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 45672 . . . 4 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3 elnelne2 3057 . . . . 5 ((𝑋𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅)
43expcom 415 . . . 4 (∅ ∉ 𝑃 → (𝑋𝑃𝑋 ≠ ∅))
52, 4syl 17 . . 3 (𝐹 Fn 𝐴 → (𝑋𝑃𝑋 ≠ ∅))
65imp 408 . 2 ((𝐹 Fn 𝐴𝑋𝑃) → 𝑋 ≠ ∅)
7 simpr 486 . . . . . 6 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → 𝑦𝑋)
8 fundcmpsurinj.h . . . . . . . 8 𝐻 = (𝑝𝑃 (𝐹𝑝))
91, 8imasetpreimafvbijlemfv 45684 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝑃𝑦𝑋) → (𝐻𝑋) = (𝐹𝑦))
1093expa 1119 . . . . . 6 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → (𝐻𝑋) = (𝐹𝑦))
117, 10jca 513 . . . . 5 (((𝐹 Fn 𝐴𝑋𝑃) ∧ 𝑦𝑋) → (𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦)))
1211ex 414 . . . 4 ((𝐹 Fn 𝐴𝑋𝑃) → (𝑦𝑋 → (𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦))))
1312eximdv 1921 . . 3 ((𝐹 Fn 𝐴𝑋𝑃) → (∃𝑦 𝑦𝑋 → ∃𝑦(𝑦𝑋 ∧ (𝐻𝑋) = (𝐹𝑦))))
14 n0 4310 . . 3 (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦𝑋)
15 df-rex 3071 . . 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 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2940  wnel 3046  wrex 3070  c0 4286  {csn 4590   cuni 4869  cmpt 5192  ccnv 5636  cima 5640   Fn wfn 6495  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508
This theorem is referenced by:  imasetpreimafvbijlemf1  45686
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