Theorem marypha2lem4 9043

Description: Lemma for marypha2 9044. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypothesis

Ref	Expression
marypha2lem.t	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))

Assertion

Ref	Expression
marypha2lem4	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑋

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		marypha2lem.t	. . . . . 6 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
2	1	marypha2lem2 9041	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3	2	imaeq1i 5915	. . . 4 ⊢ (𝑇 “ 𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋)
4		df-ima 5553	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
5	3, 4	eqtri 2762	. . 3 ⊢ (𝑇 “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
6		resopab2 5893	. . . . . 6 ⊢ (𝑋 ⊆ 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
7	6	adantl 485	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
8	7	rneqd 5796	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
9		rnopab 5812	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		df-rex 3060	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)))
11	10	bicomi 227	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥))
12	11	abbii 2804	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
13		df-iun 4896	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
14	12, 13	eqtr4i 2765	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥)
15	14	a1i 11	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
16	9, 15	syl5eq 2786	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
17	8, 16	eqtrd 2774	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
18	5, 17	syl5eq 2786	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
19		fnfun 6468	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	19	adantr 484	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → Fun 𝐹)
21		funiunfv 7050	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋))
22	20, 21	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋))
23	18, 22	eqtrd 2774	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2712  ∃wrex 3055   ⊆ wss 3857  {csn 4531  ∪ cuni 4809  ∪ ciun 4894  {copab 5105   × cxp 5538  ran crn 5541   ↾ cres 5542   “ cima 5543  Fun wfun 6363   Fn wfn 6364  ‘cfv 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-fv 6377

This theorem is referenced by:  marypha2  9044