Theorem imasetpreimafvbijlemfv1 44855

Description: Lemma for imasetpreimafvbij 44858: 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 44842	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3060	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑋 ≠ ∅)
4	3	expcom 414	. . . 4 ⊢ (∅ ∉ 𝑃 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅))
5	2, 4	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑃 → 𝑋 ≠ ∅))
6	5	imp 407	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ ∅)
7		simpr 485	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
8		fundcmpsurinj.h	. . . . . . . 8 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
9	1, 8	imasetpreimafvbijlemfv 44854	. . . . . . 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 512	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦)))
12	11	ex 413	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))))
13	12	eximdv 1920	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (∃𝑦 𝑦 ∈ 𝑋 → ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))))
14		n0 4280	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋)
15		df-rex 3070	. . 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 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943   ∉ wnel 3049  ∃wrex 3065  ∅c0 4256  {csn 4561  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  imasetpreimafvbijlemf1  44856