Theorem rnmposs 32604

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 7524	. . . 4 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	eqabri 2872	. . 3 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
4		2r19.29 3120	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶))
5		eleq1 2817	. . . . . . . 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 3180	. . . . 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 3954	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3916  ran crn 5641   ∈ cmpo 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-cnv 5648  df-dm 5650  df-rn 5651  df-oprab 7393  df-mpo 7394

This theorem is referenced by:  fedgmul  33633  raddcn  33925  br2base  34266  sxbrsiga  34287