Theorem imasetpreimafvbijlemfv1 47328

Description: Lemma for imasetpreimafvbij 47331: 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 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemfv1	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))

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

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	0nelsetpreimafv 47315	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3056	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅)
4	3	expcom 413	. . . 4 ⊢ (∅ ∉ 𝑃 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅))
5	2, 4	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅))
6	5	imp 406	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ ∅)
7		simpr 484	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
8		fundcmpsurinj.h	. . . . . . . 8 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
9	1, 8	imasetpreimafvbijlemfv 47327	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃 ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦))
10	9	3expa 1117	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦))
11	7, 10	jca 511	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))
12	11	ex 412	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))))
13	12	eximdv 1915	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (∃𝑦 𝑦 ∈ 𝑋 → ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))))
14		n0 4359	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋)
15		df-rex 3069	. . 3 ⊢ (∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))
16	13, 14, 15	3imtr4g 296	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑋 ≠ ∅ → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦)))
17	6, 16	mpd 15	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938   ∉ wnel 3044  ∃wrex 3068  ∅c0 4339  {csn 4631  ∪ cuni 4912   ↦ cmpt 5231  ^◡ccnv 5688   “ cima 5692   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-nel 3045  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:  imasetpreimafvbijlemf1  47329