Theorem fundcmpsurinjlem3 47414

Description: Lemma 3 for fundcmpsurinj 47423. (Contributed by AV, 3-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
fundcmpsurinjlem3	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))

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

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

Proof of Theorem fundcmpsurinjlem3

Step	Hyp	Ref	Expression
1		fundcmpsurinj.h	. . 3 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
2	1	a1i 11	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝)))
3		imaeq2 6043	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐹 “ 𝑝) = (𝐹 “ 𝑋))
4	3	unieqd 4896	. . 3 ⊢ (𝑝 = 𝑋 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
5	4	adantl 481	. 2 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 = 𝑋) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
6		simpr 484	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑃)
7		funimaexg 6623	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐹 “ 𝑋) ∈ V)
8	7	uniexd 7736	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → ∪ (𝐹 “ 𝑋) ∈ V)
9	2, 5, 6, 8	fvmptd 6993	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∃wrex 3060  Vcvv 3459  {csn 4601  ∪ cuni 4883   ↦ cmpt 5201  ^◡ccnv 5653   “ cima 5657  Fun wfun 6525  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539

This theorem is referenced by:  imasetpreimafvbijlemfv  47416