Theorem imasetpreimafvbijlemf 46069

Description: Lemma for imasetpreimafvbij 46074: 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 46064	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ ran 𝐹)
3		fnima 6681	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
4	3	adantr 482	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → (𝐹 “ 𝐴) = ran 𝐹)
5	2, 4	eleqtrrd 2837	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑝 ∈ 𝑃) → ∪ (𝐹 “ 𝑝) ∈ (𝐹 “ 𝐴))
6		fundcmpsurinj.h	. 2 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
7	5, 6	fmptd 7114	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  {csn 4629  ∪ cuni 4909   ↦ cmpt 5232  ^◡ccnv 5676  ran crn 5678   “ cima 5680   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544

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-sep 5300  ax-nul 5307  ax-pr 5428

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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by:  imasetpreimafvbijlemf1  46072  imasetpreimafvbijlemfo  46073