Theorem rnmposs 30417

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 7277	. . . 4 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	abeq2i 2947	. . 3 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
4		2r19.29 3333	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝐷 ∧ 𝑧 = 𝐶))
5		eleq1 2899	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝐷 ↔ 𝐶 ∈ 𝐷))
6	5	biimparc 482	. . . . . . 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 3291	. . . . 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 415	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
11	3, 10	syl5bi 244	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → (𝑧 ∈ ran 𝐹 → 𝑧 ∈ 𝐷))
12	11	ssrdv 3966	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3137  ∃wrex 3138   ⊆ wss 3929  ran crn 5549   ∈ cmpo 7151

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-cnv 5556  df-dm 5558  df-rn 5559  df-oprab 7153  df-mpo 7154

This theorem is referenced by:  fedgmul  31049  raddcn  31191  br2base  31546  sxbrsiga  31567