Theorem imasubclem1 49359

Description: Lemma for imasubc 49406. (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 7866	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → ◡𝐹 ∈ V)
4	3	imaexd 7858	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ V)
5		imasubclem1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		cnvexg 7866	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → ◡𝐺 ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → ◡𝐺 ∈ V)
8	7	imaexd 7858	. . 3 ⊢ (𝜑 → (◡𝐺 “ 𝐵) ∈ V)
9	4, 8	xpexd 7696	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵)) ∈ V)
10		fvex 6847	. . . 4 ⊢ (𝐻‘𝐶) ∈ V
11	10	imaex 7856	. . 3 ⊢ ((𝐻‘𝐶) “ 𝐷) ∈ V
12	11	rgenw 3055	. 2 ⊢ ∀𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V
13		iunexg 7907	. 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 3051  Vcvv 3440  ∪ ciun 4946   × cxp 5622  ^◡ccnv 5623   “ cima 5627  ‘cfv 6492

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 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500

This theorem is referenced by:  imasubclem2  49360  imasubclem3  49361