Theorem marypha2lem4 9322

Description: Lemma for marypha2 9323. 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 9320	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3	2	imaeq1i 6005	. . . 4 ⊢ (𝑇 “ 𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋)
4		df-ima 5627	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
5	3, 4	eqtri 2754	. . 3 ⊢ (𝑇 “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
6		resopab2 5984	. . . . . 6 ⊢ (𝑋 ⊆ 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
7	6	adantl 481	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
8	7	rneqd 5877	. . . 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 5893	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		df-rex 3057	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)))
11	10	bicomi 224	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥))
12	11	abbii 2798	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
13		df-iun 4941	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
14	12, 13	eqtr4i 2757	. . . . . 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	eqtrid 2778	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
17	8, 16	eqtrd 2766	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
18	5, 17	eqtrid 2778	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
19		fnfun 6581	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	19	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → Fun 𝐹)
21		funiunfv 7182	. . 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 2766	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056   ⊆ wss 3897  {csn 4573  ∪ cuni 4856  ∪ ciun 4939  {copab 5151   × cxp 5612  ran crn 5615   ↾ cres 5616   “ cima 5617  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  marypha2  9323