Theorem imasubclem1 49804

Description: Lemma for imasubc 49851. (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 7923	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → ◡𝐹 ∈ V)
4	3	imaexd 7915	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ V)
5		imasubclem1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		cnvexg 7923	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → ◡𝐺 ∈ V)
7	5, 6	syl 18	. . . 4 ⊢ (𝜑 → ◡𝐺 ∈ V)
8	7	imaexd 7915	. . 3 ⊢ (𝜑 → (◡𝐺 “ 𝐵) ∈ V)
9	4, 8	xpexd 7752	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵)) ∈ V)
10		fvex 6897	. . . 4 ⊢ (𝐻‘𝐶) ∈ V
11	10	imaex 7913	. . 3 ⊢ ((𝐻‘𝐶) “ 𝐷) ∈ V
12	11	rgenw 3089	. 2 ⊢ ∀𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V
13		iunexg 7962	. 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 597	1 ⊢ (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085  Vcvv 3463  ∪ ciun 4960   × cxp 5662  ^◡ccnv 5663   “ cima 5667  ‘cfv 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-xp 5670  df-rel 5671  df-cnv 5672  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fv 6547

This theorem is referenced by:  imasubclem2  49805  imasubclem3  49806