Theorem elimampo 7547

Description: Membership in the image of an operation. (Contributed by SN, 27-Apr-2025.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
elimampo.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
elimampo.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
elimampo.y	⊢ (𝜑 → 𝑌 ⊆ 𝐵)

Assertion

Ref	Expression
elimampo	⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))

Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elimampo

Step	Hyp	Ref	Expression
1		df-ima 5674	. . . 4 ⊢ (𝐹 “ (𝑋 × 𝑌)) = ran (𝐹 ↾ (𝑋 × 𝑌))
2	1	eleq2i 2855	. . 3 ⊢ (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)))
3		rngop.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	reseq1i 5974	. . . . . 6 ⊢ (𝐹 ↾ (𝑋 × 𝑌)) = ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌))
5		elimampo.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
6		elimampo.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐵)
7		resmpo 7530	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
8	5, 6, 7	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
9	4, 8	eqtrid 2810	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
10	9	rneqd 5928	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (𝑋 × 𝑌)) = ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
11	10	eleq2d 2849	. . 3 ⊢ (𝜑 → (𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)))
12	2, 11	bitrid 286	. 2 ⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)))
13		elimampo.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
14		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
15	14	elrnmpog 7545	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
16	13, 15	syl 18	. 2 ⊢ (𝜑 → (𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
17	12, 16	bitrd 282	1 ⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ⊆ wss 3905   × cxp 5659  ran crn 5662   ↾ cres 5663   “ cima 5664   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  psdmul  22329  elrgspnlem2  33563