Theorem fundcmpsurinjlem3 48007

Description: Lemma 3 for fundcmpsurinj 48016. (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 6046	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐹 “ 𝑝) = (𝐹 “ 𝑋))
4	3	unieqd 4879	. . 3 ⊢ (𝑝 = 𝑋 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
5	4	adantl 485	. 2 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 = 𝑋) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
6		simpr 488	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑃)
7		funimaexg 6609	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐹 “ 𝑋) ∈ V)
8	7	uniexd 7726	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → ∪ (𝐹 “ 𝑋) ∈ V)
9	2, 5, 6, 8	fvmptd 6984	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087  Vcvv 3455  {csn 4583  ∪ cuni 4866   ↦ cmpt 5182  ^◡ccnv 5647   “ cima 5651  Fun wfun 6516  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fv 6530

This theorem is referenced by:  imasetpreimafvbijlemfv  48009