Theorem imasubclem1 49291

Description: Lemma for imasubc 49338. (Contributed by Zhi Wang, 6-Nov-2025.)

Hypotheses

Ref	Expression
imasubclem1.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
imasubclem1.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)

Assertion

Ref	Expression
imasubclem1	⊢ (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐻(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem imasubclem1

Step	Hyp	Ref	Expression
1		imasubclem1.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		cnvexg 7864	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → ◡𝐹 ∈ V)
4	3	imaexd 7856	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ V)
5		imasubclem1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		cnvexg 7864	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → ◡𝐺 ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → ◡𝐺 ∈ V)
8	7	imaexd 7856	. . 3 ⊢ (𝜑 → (◡𝐺 “ 𝐵) ∈ V)
9	4, 8	xpexd 7694	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵)) ∈ V)
10		fvex 6845	. . . 4 ⊢ (𝐻‘𝐶) ∈ V
11	10	imaex 7854	. . 3 ⊢ ((𝐻‘𝐶) “ 𝐷) ∈ V
12	11	rgenw 3053	. 2 ⊢ ∀𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V
13		iunexg 7905	. 2 ⊢ ((((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵)) ∈ V ∧ ∀𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V) → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)
14	9, 12, 13	sylancl 586	1 ⊢ (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  ∪ ciun 4944   × cxp 5620  ^◡ccnv 5621   “ cima 5625  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498

This theorem is referenced by:  imasubclem2  49292  imasubclem3  49293