Theorem iunrnmptss 32501

Description: A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Hypotheses

Ref	Expression
iunrnmptss.1	⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
iunrnmptss.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
iunrnmptss	⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)

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

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

Proof of Theorem iunrnmptss

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3055	. . . 4 ⊢ (∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶 ↔ ∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶))
2		iunrnmptss.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	2	ralrimiva 3126	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
4		eqid 2730	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	elrnmptg 5928	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
7	6	anbi1d 631	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
8	7	exbidv 1921	. . . . 5 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		r19.41v 3168	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶))
10		iunrnmptss.1	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
11	10	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
12	11	biimpa 476	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐷)
13	12	reximi 3068	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
14	9, 13	sylbir 235	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
15	14	exlimiv 1930	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
16	8, 15	biimtrdi 253	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
17	1, 16	biimtrid 242	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
18	17	ss2abdv 4032	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶} ⊆ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷})
19		df-iun 4960	. 2 ⊢ ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶}
20		df-iun 4960	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐷 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷}
21	18, 19, 20	3sstr4g 4003	1 ⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ∪ ciun 4958   ↦ cmpt 5191  ran crn 5642

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 5254  ax-nul 5264  ax-pr 5390

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 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-cnv 5649  df-dm 5651  df-rn 5652

This theorem is referenced by:  fnpreimac  32602