Theorem fundcmpsurinjlem3 48627

Description: Lemma 3 for fundcmpsurinj 48636. (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 6054	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐹 “ 𝑝) = (𝐹 “ 𝑋))
4	3	unieqd 4883	. . 3 ⊢ (𝑝 = 𝑋 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
5	4	adantl 487	. 2 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 = 𝑋) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑋))
6		simpr 490	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑃)
7		funimaexg 6637	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐹 “ 𝑋) ∈ V)
8	7	uniexd 7769	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → ∪ (𝐹 “ 𝑋) ∈ V)
9	2, 5, 6, 8	fvmptd 7012	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∃wrex 3090  Vcvv 3454  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  Fun wfun 6541  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fv 6555

This theorem is used by:  imasetpreimafvbijlemfv  48629