Theorem imasetpreimafvbijlemf 47884

Description: Lemma for imasetpreimafvbij 47889: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-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
imasetpreimafvbijlemf	⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemf

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	uniimaelsetpreimafv 47879	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ ran 𝐹)
3		fnima 6616	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
4	3	adantr 481	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → (𝐹 “ 𝐴) = ran 𝐹)
5	2, 4	eleqtrrd 2842	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ (𝐹 “ 𝐴))
6		fundcmpsurinj.h	. 2 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
7	5, 6	fmptd 7056	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063  {csn 4556  ∪ cuni 4839   ↦ cmpt 5154  ^◡ccnv 5618  ran crn 5620   “ cima 5622   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494

This theorem is referenced by:  imasetpreimafvbijlemf1  47887  imasetpreimafvbijlemfo  47888