Theorem fundcmpsurinjlem3 47401

Description: Lemma 3 for fundcmpsurinj 47410. (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 6027	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐹 “ 𝑝) = (𝐹 “ 𝑋))
4	3	unieqd 4884	. . 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 6603	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐹 “ 𝑋) ∈ V)
8	7	uniexd 7718	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → ∪ (𝐹 “ 𝑋) ∈ V)
9	2, 5, 6, 8	fvmptd 6975	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3447  {csn 4589  ∪ cuni 4871   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641  Fun wfun 6505  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  imasetpreimafvbijlemfv  47403