Theorem rnmposs 32691

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 7566	. . . 4 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	eqabri 2883	. . 3 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
4		2r19.29 3137	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶))
5		eleq1 2827	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝐷 ↔ 𝐶 ∈ 𝐷))
6	5	biimparc 479	. . . . . . 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 3199	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶) → 𝑧 ∈ 𝐷)
9	4, 8	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → 𝑧 ∈ 𝐷)
10	9	ex 412	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
11	3, 10	biimtrid 242	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ 𝐷))
12	11	ssrdv 4001	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ran crn 5690   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  fedgmul  33659  raddcn  33890  br2base  34251  sxbrsiga  34272