Theorem imasetpreimafvbijlemfv1 47878

Description: Lemma for imasetpreimafvbij 47881: 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 47865	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3049	. . . . 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 47877	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃 ∧ 𝑦 ∈ 𝑋) → (𝐻‘𝑋) = (𝐹‘𝑦))
10	9	3expa 1119	. . . . . 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 1919	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → (∃𝑦 𝑦 ∈ 𝑋 → ∃𝑦(𝑦 ∈ 𝑋 ∧ (𝐻‘𝑋) = (𝐹‘𝑦))))
14		n0 4294	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋)
15		df-rex 3063	. . 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933   ∉ wnel 3037  ∃wrex 3062  ∅c0 4274  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5624   “ cima 5628   Fn wfn 6488  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by:  imasetpreimafvbijlemf1  47879