Theorem rnmposs 33154

Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)

Hypothesis

Ref	Expression
rnmposs.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
rnmposs	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   𝐹(𝑥, 𝑦)

Proof of Theorem rnmposs

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnmposs.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	rnmpo 7550	. . . 4 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	eqabri 2904	. . 3 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
4		2r19.29 3150	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶))
5		eleq1 2850	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝐷 ↔ 𝐶 ∈ 𝐷))
6	5	biimparc 485	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶) → 𝑧 ∈ 𝐷)
7	6	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶) → 𝑧 ∈ 𝐷))
8	7	rexlimivv 3206	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶) → 𝑧 ∈ 𝐷)
9	4, 8	syl 18	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → 𝑧 ∈ 𝐷)
10	9	ex 418	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
11	3, 10	biimtrid 245	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ 𝐷))
12	11	ssrdv 3940	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  ran crn 5660   ∈ cmpo 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-oprab 7421  df-mpo 7422

This theorem is used by:  fedgmul  34149  raddcn  34447  br2base  34788  sxbrsiga  34809