Theorem marypha2lem4 9409

Description: Lemma for marypha2 9410. 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 9407	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
3	2	imaeq1i 6053	. . . 4 ⊢ (𝑇 “ 𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋)
4		df-ima 5668	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
5	3, 4	eqtri 2783	. . 3 ⊢ (𝑇 “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋)
6		resopab2 6032	. . . . . 6 ⊢ (𝑋 ⊆ 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
7	6	adantl 487	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))})
8	7	rneqd 5922	. . . 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 5938	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		df-rex 3087	. . . . . . . . 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 2827	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
13		df-iun 4953	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)}
14	12, 13	eqtr4i 2786	. . . . . 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 2807	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
17	8, 16	eqtrd 2795	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
18	5, 17	eqtrid 2807	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥))
19		fnfun 6633	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	19	adantr 486	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → Fun 𝐹)
21		funiunfv 7246	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋))
22	20, 21	syl 18	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋))
23	18, 22	eqtrd 2795	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086   ⊆ wss 3899  {csn 4584  ∪ cuni 4867  ∪ ciun 4951  {copab 5167   × cxp 5653  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  marypha2  9410